Part - I. Mathematics

Transcription

1 Part - I Mathematics

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3 CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet of a Set.6 Uio of Sets.6. Properties of Uio Operatio.7 Itersectio of Sets.7. Disjoit Sets.7. Properties of Itersectio Operatio.7. Relative Complemet of a Set.7.4 Illustrative Example.8 De Morga's Laws.9 Distributive Laws of Uio ad Itersectio. Illustrative Examples. Summary. Check your Progress - Aswers. Questios for Self - Study.4 Suggested Readigs. OBJECTIVES After studyig the cocept of a set ad its fudametal operatios you ca explai the followig : Studets ca verify : Cocept of a set Complemet of a set Uio ad itersectio of sets. De Morga's Laws. Distributive laws of uio ad itersectio. Studets ca solve problems ivolvig umerical data.. INTRODUCTION The theory of sets forms the basis of the moder mathematics. The begiig of the theory ca be traced from the Germa mathematicia Cator at the ed of the 8th cetury. At preset it covers a very extesive part of mathematics. However, we restrict ourselves to a very elemetary part of it.. SET - CONCEPT I our everyday life we ofte make use of collective phrases. Like, (i) (ii) Studets wearig spectacles, Wagos attached to a trai, Set Theory /

4 (iii) (iv) (v) Flowerig trees i a garde, Studets i a class, A pack of cards. Such collectios are called 'sets' i mathematics. To describe a set we eed the followig cosideratios. (a) (b) set or collectio elemets i a set (c) rule of property which eables us to say whether a give object is i a set or ot. e.g. I a set "studets wearig spectacles" we ote that a elemet is "every studet who wears spectacles" ad the rule is "oly those studets who wear spectacles.".. Sets ad Elemets Cosider a set cosistig of umbers,,, 4, 5. Each of these umbers is a elemet of the set. We deote this set by a letter A. A = {,,, 4, 5} The fact that a umber is i a set A is writte as " belogs to the set A". This cocept of belogig to the set is symbolically deoted as A ad read as " belogs to A". While the fact that the elemet 6 is ot i the set A; i.e. 6 does ot belog to the set A is symbolically writte as 6 A ad read as "6 does ot belog to A". There are various ways of represetig a set. Oe such method is idicated above to describe a set. Aother method is to state all the elemets of a set. There is oe more method, which we shall study ow. The set is described by usig the property of the elemets. e.g. The fact that the elemet x of the set A havig a property P is described as P(x) ad the set is writte as A = {x P(x)} The above set A = {,,, 4, 5} ca ow be writte as i this otatio as A = {x x is a positive iteger betwee ad 6} A elemet of the set is couted oce oly, i.e. {,,, } is the same as {,, }. Also set is regarded as the same eve if its elemets are writte i differet order. e.g. {p, q, r, s} = {r, p, s, q} = {s, r, p, q}. Defiitio : Two sets are said to be equal if they cotai the same elemets. e.g. A = {,4}, B = { x x 6x + 8 = } The elemets of the set B are the roots of the equatio x 6x + 8 =. We kow from Elemetary Algebra that the roots of this equatio are ad 4. Thus by the defiitio two sets A ad B are equal; ad we write A = B. Defiitio : A set havig oly oe elemet is called a sigleto set. e.g. {prime miister of Idia}, {a}, {}, {b} are. all sigleto sets of : prime miister of Idia, a,, b. Next, cosider a set { x x =, x = }. Mathematics & Statistics /

5 We kow that there is o umber which is equal to ad at the same time. Thus this set has o elemet. Such a set havig o elemet is called a empty set. Defiitio : A set havig o elemet is called a empty (or ull or void) set ad is deoted by. Thus = {x x x} All empty sets are equal.. Check your progress.. Use appropriate symbols i the blak. (a).. {,,,4} (b) 5.. {,6,7,8}. Are the followig sets equal? A = {x x is a positive iteger, 5} B = {,,, 4, 5} C = {x x is a root of the equatio x x + = }. State whether true or false. A = {x x is a positive iteger, 4}, B = ad A = B. C = (x x is a root of the equatio 4x + = }, ad C is a sigleto set. Cosider two sets.. SUBSET A = {x x is a studet i st year class} B = {y y is a studet wearig spectacles i st year class} All studets who wear spectacles are i a st year class. I other words every y is x. Obviously every studet i the class may ot wear spectacles i.e. every x is ot y. i.e. Every elemet of the set B is a elemet of A, but every elemet of A is ot a elemet of B. I this case, we say that B is a subset of A We itroduce the followig otatio. If P ad Q are two statemets such that, if P is true the Q must be true; we say that P implies Q. We express this i a symbolic form as P Q. (read as P implies Q) e.g. (x = ) ( x 6x + 9 = ) If i additio to P Q, we have Q P, we have both sided implicatios, ad we write it as P Q. read as "P implies ad is implied by Q." e.g. x = x =8, whe x is real. For the two sets C, D defied as C = {x x is a triagle} D = {x x is a equilateral triagle} We see that x D x C. Defiitio : B is called a subset of A if x B x A. The statemet that B is a subset of A is symbolically writte as BA. If x A x B, the A is a subset of B. i.e. A B Set Theory /

6 ad is read as "A is cotaied i B" or "B cotais A" or "A is a subset of B". To illustrate the relatioship betwee gets, we use differet diagrams called Ve diagrams... Proper ad Improper Subsets We shall ow cosider two extreme cases of subsets. I the illustratio of sets A = {x x is a studet i st year class} B = {y y is a studet wearig spectacle i st year class} (i) It may happe that o studet i st year class wears spectacles. i.e. B = Hece ca be regarded as a subset of A. I fact A for every set A. (ii) Aother possibility is, every studet i st year class may wear spectacles. i.e. B = A. I this case B A A A. I fact every set A ca be regarded as a subset of itself. Both ad A are called improper subsets of A. Except ad A all other subsets of A are called proper subsets of A. e.g. Let A = {a, b, c} The possible subsets of A are, A = {a}, A = {b}, A = {c}, A 4 = {a, b}, A 5 = {b. c}, A 6 = {c, a), A = a, b, c} Thus A, A, A, A 4, A 5, A 6 are proper subsets of A. The set of all subsets of A is called a power set of A ad is deoted by P(A). The total umber of elemets of P(A) is give by a formula : (P(A)) = (A) where (A) deote the umber of elemets i the set A. Here (A) = ad ( P (A) ) = = 8. We easily cout the umber subsets of A as 8. Next cosider, S = {,,,5,7}, T = {,5,7} Here T is a proper subset of S. T S. Sometimes S is called superset of T... Equality of Sets The equality of two sets ca be defied as : Mathematics & Statistics / 4

7 Defiitio : Two sets A ad B are said to be equal if each of A ad B is a subset of other. A = B A B ad B A. This ca be proved as follows : Let A B ad B A. x A x B ad x B x A. As all these relatios hold at the same time, it is obvious that A ad B have the same elemets. i.e. A = B. Coversely, let us suppose that A = B. Hece x A x B i.e. A B ad x B x A i.e. B A. This provides us with a very importat tool to prove the equality of two sets (especially whe they cotai ifiite umber of elemets).. Trasitivity of set iclusio Now cosider, A = {p, q, r, s} B = {o, p, q, r, s, t} C = {o, p, q, r, s, t, u, v} We clearly see the followig relatios. i.e. A B, B C. Also A C. (AB, BC) AC. This property kow as set iclusio is said to be trasitive. The geeral proof of the above property is give below. A B B C meas x A x B meas x B x C Hece A B ad B C meas i.e. x A x B x C x A x C i.e. A C. We have the followig ve diagram to illustrate the above two properties.. Check your progress.. State whether true or false. (a) {,,} (b) {, 4, 6} Set Theory / 5

8 (c) {, } {x x is a positive iteger, x } (d) ad A are improper subsets of A. (e) The proper subsets of A = {, 5, 6} are A = {}, A = {5}, A = {6}, A 4 = {, 5}, A 5 = {, 6}, A 6 = {5, 6}.. Write dow all possible subsets of the set A = {,, }.. (a) Is A = {a, b, c} a super set of B = {a, b, c, d}? (b) Write dow all possible subsets of B = {p, q, r, s} each cotaiig two - elemets oly. Cosider the followig sets..4 UNIVERSAL SET A = {x x is a studet i st year class} B = {x x is a studet i a college i Idia} C = {x x is a studet i a college i the world} U = {x x is a studet} We see that A B,B C, C U. I fact A, B, C are all subsets of a fixed set U. We may fid umber of sets which are subsets of a fixed set like U. This fixed set is called the uiversal set. Let P, Q, R, S, T,... be the set of books writte i Marathi, Gujarathi, Begali, Hidi Frech, Germa, laguages. All these are subsets of "a set of books' say X. I this case X is the uiversal set. May other illustratios of uiversal sets ca be costructed..5 COMPLEMENT OF A SET Cosider a uiversal set X = {x x is a tree} ad a set A = {x x is a tree havig height more tha meters}. We ca form aother set B = {x x is a tree havig height less tha meters}. We observe that sets A ad B together form a uiversal set X. I this case, B is called complemet of the set A. The elemets of B are the elemets of X but are ot elemets of A. The complemet of a set A is deoted by A. Defiitio: The complemet of a set A has elemets i X which are ot i A. i.e. A = {x x X, x A} e.g. Let X = {,,,, 4, 5} ad A = {,, 4} The A = {,, 5} The complemet of A is {,, 4}, which is A agai. (A) = A This ca be proved as follows : x (A) xa xa i.e. (A')' = A We shall represet X, A, A' by the followig ve diagram. Mathematics & Statistics / 6

9 We ote the followig poits regardig complemet of a set. (i) ' = X (ii) X' = (iii) If A B the B' A' We shall prove it as: Let x B' x B, But A B x A x A' Thus x B' x A' B' A'. The represetatio by a ve diagram is as :.4 &.5 Check your progress: Solve the Followig Give a uiversal set X = {,,, 4, 5, 6, 7, 8, 9} Obtai the complemet of the followig sets usig uiversal X (above) (a) A = {,, 5, 7, 9} (b) B = {, 4, 6, 8} (c) C = {,, 9} (d) D = {,,,4, 5,6,7,8,9} (e).6 UNION OF SETS We shall ow begi with operatios o sets or what is usually called as Algebra of sets. Let A = {a, b, c, d, e}, B = {a, c, f, g, h} ad C = {a, b, c, d, e, f, g, h} We fid here that every elemet of A is a elemet of C ad every elemet of B is a elemet of C. I other words, if a elemet belogs to A, it belogs to C ad similarly if a elemet belogs to B, it belogs to C. i.e. A elemet of C is either a elemet of A or a elemet of B or it is a commo elemet of both A ad B. Such a set Set Theory / 7

10 C is called uio of two sets A ad B. This set is deoted by A B, ad is read as "A uio B". Defiitio: The uio of two sets A ad B is A B = {x xa or xb or x both A ad B } or A B = {x x A ad or x B} Thus A B is the set of all elemets of A ad B, the commo elemets, if ay, beig take oce oly. The followig Ve-diagram will illustrate the statemet..6. Properties of Uio Operatio We shall state some properties of the uio operatio (i) AB = BA (commutativity) Proof.: xab x A ad or xb x B ad or x A x BA i.e. we get the same set, i whatever order the sets are take for uio. I this case we say that uio is commutative. (ii) (iii) (iv) (v) (vi) A = A A X = X A A' = X Proof.: A A = A Proof.: AAB ad BAB x A A' xa ad or x A' x A ad x A' x X x A A xa ad or x A xa Both these follow from the Ve diagram of A B. (Fig..8) (vii) If A B, the A B = B Proof.: Sice A B, x A x B X A B x A ad or x B x B (i either case x B) Mathematics & Statistics / 8

11 Similarly, we ca show that BAB A B B A B = B. (viii) A (BC) = (AB) C (associativity) Proof. : x A ( B C ) x A ad or x (BC) x A ad or x B ad or x C x (A B) ad or x C x (A B) C I this case we say that uio operatio is associative. i.e. if we remove the brackets o either side we get the same result, (ix) viz. A B C. A B C = {x x A ad or x B ad or x C} = {x x at least oe of A, B, C} If AC ad B C the (AB) C C Proof.: A C meas x A x C B C meas x B x C x (AB) xa ad or x B x C (A B) C. Here we say that uio operatio is trasitive. (x) Priciple of iclusio ad exclusio. Let A ad B be fiite subsets of a uiversal set. If A ad B are disjoit sets ( i.e. A B = ) Set Theory / 9

12 ( A B) = ( A ) + ( B ) I particular (A) + ( A') = (X).6 Check your progress: Give a uiversal set X = {5,, 5,, 5,, 5} A = {,, }, B = {5,, 5, 5}, C = {}. Write dow the followig sets (a) A B (b) B C (c) A C (d) A' B' (e) A' C'. State whether true or false. (a) C B (b) A C = A (c) B C = B (d) A B A C. Verify the associative law A (B C) = (A B) C.7 INTERSECTION OF SETS This is aother operatio with sets. Let A = {,,,4,6} B = {4, 6, 8, } C = {4, 6} We fid here that every elemet of C is a elemet of A as well as a elemet of B. I other words, if a elemet belogs to C; it belogs to both A ad B. i.e. a elemet of C is commo to both A ad B. Such a set C is called itersectio of two sets A ad B. This set is deoted by A B; read as "A itersectio B". Defiitio: The itersectio of two sets A ad B is A B = {x x A ad x B} Mathematics & Statistics /

13 Thus A B is a set of all commo elemets of A ad B. This has bee show above i Fig Disjoit Sets Now cosider the two sets, A = {x x is a eve iteger} B = {x x is a odd iteger) We kow that there is o iteger which is both eve ad odd at the same time. Thus these two sets A ad B have o commo elemet. Their itersectio is a empty set. A B = Two such sets are called disjoit sets. Defiitio : Two sets are disjoit, if they have o elemet i commo. e.g. A ad A' are always disjoit. A A' = for ay set A..7. Properties of Itersectio Operatio We shall list some properties of itersectio. (i) A B = B A (commutativity) Proof.: xab xa ad x B x B ad x A x ( B A ) i.e. A B = B A We say that itersectio operatio is commutative. i.e. We get the same set i whatever order the sets are take for itersectio. (ii) A = (iii) (iv) (v) A X = A A A' = A A = A Proof.: x A A x A ad x A x A A A = A Set Theory /

14 (vi) We ca easily see from Ve diagram of Ve diagram represetatio is as follows: A B that (a) (A B) A (b) (A B) B (vii) If A B the A B = A Proof.: Sice A B, x A x B x A B x A ad x B x A i.e. A B = A. (i either case) (viii) A ( B C ) = ( A B ) C (associativity) Proof.: x A (B C) x A ad x ( B C) x A ad x B ad x C (x A ad x B ) ad x C x (AB) ad x C x (AB)C We say that itersectio operatio is associative. i.e. if we remove the brackets o either side we get the same result. viz. A B C = {x x A ad x B ad x C} = {x x all the sets A, B, C}. Mathematics & Statistics /

15 (ix) If C A ad C B the C ( A B ) Proof.: C A meas x C x A C B meas x C x B x C x A ad x B x ( A B). C ( A B )..7. Relative Complemet of a Set We have see the cocept of a complemet of a set i relatio to a uiversal set. Now we shall cosider complemet of a set A relative to aother set B. Defiitio: Relative complemet of A with respect to B is defied as A B' ad is usually deoted by A B. A B = {x x A ad x B} i.e. A B is a set of all elemets of A except the commo elemets of A ad B. A B is also called the differece of two sets A ad B. Similarly, relative complemet of B with respect to A is B A' = B A. B A = {x x B ad x A} i.e. B A is a set of all elemets of B except the commo elemets of A ad B. The Ve-diagram represetatio i as : (ix) Icidetally we observe the followig from the above Ve diagram Fig..7. (a) sets A B, B A ad A B are pair wise disjoits sets. i.e. every pair of sets A B, B A ad AB have o elemet is commo. (b) i.e. A B is uio of disjoit sets viz A B, B A, AB. A B = (A B) (B A) (AB) e.g. Let A = {, 4, 6, 8,, } B = {4, 8,, 6} We have A B = {, 6, } B A = {6} A B = {8,4, } Set Theory /

16 (x) AB = {,4,6,8,,, 6} = {,6, } {6} {4, 8, }. If X is the uiversal set ad A is ay of its subset the A' = X A. Geeralized Priciple of Iclusio ad Exclusio Se shall state various results without proof. The utility of them will be see i illustrative examples. (i) (ii) Let A, B, C be ay fiite subsets of a uiversal set. We have the followig rules. (AB) = (A) + (B) (AB) (ABC) = (A) + (B) + (C) (AB) (BC) (AC) + (ABC) (iii) From the Ve diagram of AB, we have the followig: (A B) = (A) (AB) = (AB') (iv) (B A ) = (B ) (A B ) = (A' B ) (v) Similarly from the Ve diagram of ABC. it follows that Number of elemets of A oly = (A) (AB) (AC) + (ABC)..7.4 Illustrative Example We shall illustrate the various operatios that we have studied so far by meas of a example. Let X= {,,,, 4, 5, 6, 7, 8, 9} A= {,, 4, 6, 8} B= {,,7,8,9} C= {,5,7,9} We have the followig : (i) AB= {,,,, 4, 6, 7, 8, 9} =X {5} (ii) B C= {,,, 5, 7, 8, 9} (iii) A C= {,,4, 5, 6,7,8,9} (iv) A B= {8} (v) B C= {7,9} (vi) A C= {} (vii) A'= X A = {,, 5, 7, 9} (viii) B'= X B = {,, 4, 5, 6} (ix) C'= X C = {,,, 4, 6, 8} Mathematics & Statistics / 4

17 (x) (xi) (xii) A B= {,, 4, 6} = A (AB) B C= {,, 8} = B (BC) C A= {5, 7, 9} =C (CA) (xiii) A' B = {5} (xiv) (xv) A B C = {,,,, 4, 5, 6, 7, 8, 9} = X A B C = (xvi) B A = {,,7, 9} (xvii) Symmetric differece A B = (A B ) (B A) = {,,,, 4, 5, 6, 7, 9} The Ve-diagram represetatio of some of these operatios is give below..7 Check your progress: Solve the followig Give X = {a, b, c, d, e, f, g, h, i} A = {a, c, d, f, g}, B = {c, d, e, f, h}, C = {a, i}. Write dow the followig sets. (i) A B (ii) B C (iii) A C (v) A' B (iv) A' B' (vi) B' A (vii) Symmetric differece A B. Draw Ve-diagram to represet the followig sets. (i) A B (ii) A B (iii) A C (v) A B (iv) A'C Set Theory / 5

18 .8 DE MORGAN'S LAWS We shall study two results kow as De Morga's laws. We shall offer theoretical proofs ad their verificatios for certai sets. (I) (A B)' = A'B' Proof.: x(ab)' x(ab) x A ad/or x B xa' ad x B' x (A'B') (AB)' = A'B' (II) (A B)' = A'B' Proof. : x (A'B') x A' ad/ o r x B' x A ad xb x (AB) x(ab)' (AB)' =A'B' We shall verify these laws for the followig sets. X = {x x is a positive iteger ad x } A = {,, 7}, B = {, 4, 6, 8, } We have AB = {} AB = {,, 4, 6, 7, 8, } A' = {, 4, 5, 6, 8, 9, } B' = {,,5,7,9} (AB)' = {,,4,5,6,7,8,9,} (AB)' = {,5,9} A'B' = {,5,9} A'B' = {,, 4, 5, 6, 7, 8, 9, } (AB)' = {, 5, 9} = A'B' Ad (AB)' = {,, 4, 5, 6, 7, 8, 9, } = A'B' This completes the verificatio. Check your progress -.8 X = {a, b, c, d, p, q, r, s} A = {a, c, d, p, s} B = {b, c, d, p, q} verify: (i) (AB )' = AB' (ii) ( A B )' = A'B'.9 DISTRIBUTIVE LAWS OF UNION AND INTERSECTION We are familiar with the followig simple operatios. (4 + 6) = (4) + ( 6) The two operatios used are multiplicatio ad additio. The above relatio is expressed as "multiplicatio is distributive over additio." I the above illustratio, we ca ot iterchage multiplicatio ad additio. + (46) ( + 4) ( + 6) The laws for uio ad itersectio are as: (I) A (B C) = (A B) (A C) By iterchagig ad i (I), we get aother distributive law: Mathematics & Statistics / 6

19 (II) A(BC) = (AB)(AC) We shall verify these laws by the followig Ve diagram: Let Hece Ad A = {I, II, III, IV} B = {II, III, VI, VII} C = {III, IV, V, VI} AB = {I, II, III, IV, VI, VII} AC = {I, II, III, IV, V, VI} BC= {II, III, IV, V, VI, VII} AB = {II, III} A C = {III, IV} B C = {III, VI} A(BC) = {I, II, III, IV, VI} = (AB) (AC) A(BC) = {II, III, IV} = (AB)(AC). Check your progress.9 For the sets A = {a, b, c, d}, B = {c, d, e, f}, C = {a, d, f, g} Verify: (i) A(BC) = (AB)(AC) (ii) A(BC) = (AB)(AC). ILLUSTRATIVE EXAMPLES () Show that each of the followig coditio is equivalet to A B. (i)ab = A (ii)ab=b (iii)b'a' (iv)ba' = X (v)ab' = Solutio: (i) A B = A meas all the elemets that are commo to A ad B are i A. i.e. all the elemets i A are elemets i B while all elemets i B are ot i A. i.e. AB. (ii) We kow that A (A B) for ay two sets A ad B. (iii) B A' AB. i.e. x B' x A' i.e. x B x A i.e. x A x B Set Theory / 7

20 i.e. AB (iv) We kow that for ay set B, B B'= X We are give that B A' = X B'A' ad hece from (iii), it follows that A B. (v) We kow that for ay set A, AA'= We are give that A B' = B'A' ad the result follows from (iii). () Prove that for ay two set A ad B (AB)A(AB). Solutio : x(ab)xa ad xb I particular let x A x (A B ) x A. i.e. (AB) A...(i) Further, if x A, the x A ad/or x B. i.e. x(ab)...(ii) From (i) ad (ii), it follows that (AB)A(AB). () Give A = {a, b, c, d, e, f}, B = {d, e, f, g, h, i} C = {b, e, h, i}, D = {d, e}, E = {b, d}, F = {b}. Let x be a ukow set. Determie which sets A, B, C, D, E or F ca be equal to X, if we are give the followig iformatio. (i) X A ad X B (ii) X B ad X C (iii) X A ad X C (iv) X B ad X C. Solutio : (i) The set which is subset of A ad B both, is A B. Here this set is D. (ii) (iii) X = D. The set is ot a subset of B, but it is a subset of C. The oly elemet of C which is ot cotaied i B is b. Hece X = F. The set is ot a subset of A ad C. The oly set which satisfies the coditio is B. X= B. (iv) The set is a subset of B, but it is ot a subset of C. Hece the set may be B or D. X = B, D. (4) Prove the followig : (i) If A B = X, the A' B (ii) A (A'B) = AB. Solutio : Mathematics & Statistics / 8

21 (i) We kow that X A' = A' ad we are give that A B = X. (ii) (A B ) A' = A' By distributive law, we get (AA')(BA') = A' i.e. (BA'). = A' i.e. B A' = A' i.e. A' B By distributive law, we have A (A'B) = (A A') (AB) = (AB) = AB A(A'B) = AB. (5) The studets i a hostel were asked whether they had a TV set or a computer i their rooms. The result showed that 65 studets had a TV set, 5 did ot have a TV set, 75 had a computer ad 5 had either a TV set or a computer. Fid the umber of studets who, (a) live i the hostel. (b) have both a TV set ad a computer. (c) have oly a computer. Solutio : We shall draw a Ve-diagram,. Let C = set of studets havig a computer, T = set of studets havig a TV set, X = set of studets who live i the hostel. Let x be the umber of studets havig TV ad computer both. By data (CT) = x (T) = 65 (C) = 75 (CT)' = 5 Also there are 5 studets who do ot have a TV set. 75 x + 5 = 5 i.e. 5 5 = x i.e. x = 75. i.e. umber of studets havig both TV & computer is 75. (b) Thus the umber of studets havig both TV set ad a computer is 75. (a) Total umber of studets stayig i the hostel = (X) = (CT) + (CT)' = (C) + (T) (CT) + 5 Set Theory / 9

22 (c) = = 8. Number of studets havig oly a computer = (C) (CT) = =. (6) Amog studets, study Mathematics, study Physics, 45 study Biology. 5 study Mathematics ad Biology. 7 study Mathematics ad Physics. study Physics ad Biology. do ot study ay of the three subjects. The fid, (a) umber of studets who study all the three subjects. (b) umber of studets who study Mathematics oly. Solutio : Here also we ca draw a Ve-diagram to represet the data. Let X = set of studets P = set of studets who study Physics M = set of studets who study Mathematics B = set of studets who study Biology x = umber of studets who study all the three subjects. By data (P) = (M) = (B) = 45 (PM) = 7 (MB) = 5 (PB) = (PMB)'= (X) = ad (P M B ) = x. Number of studets who are studyig at least oe of the subjects = (PMB). (PMB) = (P) + (M) + (B) (PM) (M B ) (P B ) + (P M B) i.e. (X) (PMB)'= x i.e. = 97 + x i.e. x = 5. (a) Number of studets who study all the three subjects is 5. (b) Number of studets who study Mathematics oly = (M) (MP) (MB)+(PMB) = = 5. Mathematics & Statistics /

23 a) AB b) AB c) A ', B' d) (AB)' = B' A' e) A exits f) (AB)' = A' B' g) (AB)' = A' B'.SUMMARY A set is a well defied collectio of objects. There are types of sets. For ay two sets A ad B we ca write h) (AB) = ( A) + (B) i) (AB)= (A) + (B) (AB) j) (AB) = (A)+(B) Whe AB =. CHECK YOUR PROGRESS ANSWERS.. (a) {,,, 4} (b) 5 {, 6,,7 8}. A = B, A C, B C. (a) False (b) True.. (a) False (b) False (c) True (d) True (e) True.4 ad.5. Φ{}, {}, {}, {, }, {, }, {, } A.. (a) o, A B (b) {p, q}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s} (a) A' ={, 4, 6, 8} (b) B' = {,, 5, 7, 9} (c) C = {, 4, 5, 6, 7, 8} (d) (f) Φ X.6. (a) A B = {5,,, 5,, 5} (b) B C = {5,, 5,, 5} (c) A C = {,, } (d) A'B'={5, 5,, 5,, 5} (e) A'C'= {5, 5,,, 5, 5}. (a) False (b) True (c) False (d) True. Commo set {5,, 5,, 5, 5}.7. (i) A B = {c, d, e, f} (ii) B B =Φ (iii) A C = {a} (iv) A' B' = {a, b, g, h, i} (v) A' B = {b, i} (vi) B' A = {b, i} (vii) A B = {a, e, g, h} Set Theory /

24 . QUESTIONS FOR SELF STUDY Problems For Practice () 9 studets appeared for two papers i Mathematics at the first year examiatio. Exactly 74 ad 66 studets passed i papers I ad II respectively. 64 studets passed i both the papers. Draw a Ve-diagram to idicate these results ad hece or otherwise fid the umber of studets who have failed i both papers. () Amog 6 families, families have o childre, 4 families have oly boys ad have oly girls. How may families have both boys ad girls? () I a group of studets, 8 are takig a Mathematics class, 6 are takig a Chemistry class ad are takig both classes. (i) How may studets are takig either a Mathematics class or a Chemistry class? (ii) How may studets are takig either class? (4) Suppose that studets at a college take at least oe of the laguages Frech, Germa ad Russia. 65 studets study Frech, 45 study Germa ad 4 study Russio. Also studets study Frech ad Germa, 5 studets study Frech ad Russia, 5 studets study Germa ad Russia. Fid the umber of studets who study (i) all the three laguages, (ii) exactly oe laguage. (5) It is kow that at the uiversity 6% of professors play Teis, 5% play Cricket, 7% play Hockey, % play Teis ad Cricket. % play Teis ad Hockey, 4% play Cricket ad Hockey. Assumig that each professor play at least oe of the games, determier % of professors playig all the three games. (6) A computer compay must hire 5 programmers to hadle systems programmig tasks ad 4 programmers for applicatios programmig. Of those hired 5 will be expected to perform tasks of each type. How may programmers must be hired?.4 SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Pre- degree Mathematics by Vaze, Gosavi Mathematics & Statistics /

25 Set Theory / NOTES

26 NOTES Mathematics & Statistics / 4

27 CHAPTER Fuctios. Objectives. Itroductio. Number System.. Basic Operatios i Mathematics.. Divisibility Test. Prelimiary Cocepts.4 Correspodece.5 Fuctios.6 Types of Fuctios.7 Graph of Fuctio.8 Summary.9 Check your Progress- Aswers. Questios for Self - Study. Suggested Readigs. OBJECTIVES After studyig the umber system ad certai prelimiary cocepts, studets ca explai the followig * Fuctios i various otatios * Types of fuctios * Fuctios of fuctios * Graph of fuctios * Formula of a fuctio * Fuctio as a correspodece * Studets ca solve problems ivolvig the all above cocepts.. INTRODUCTION There is o permaet place i the world for ugly mathematics. It may be very hard to defie mathematical beauty but that is just as true of beauty of ay kid - G.H. Hardy The cocept of term relatio i mathematics has bee draw from the meaig o relatio i Eglish laguage. Accordigly two objects or quatities are related if there is a coectio or lik betwee the two objects or quatities.. NUMBER SYSTEM N = Set of all atural Numbers. = {,,,..} W = Set of whole umbers. = {,,,,..} I = Set of all Itegers. = {. -, -, -,,,,,..} Fuctios / 5

28 Q = Set of ratioal Numbers p = p, q,, I, q q.. Basic operatios i Mathematics ) Additio i) a, b are two umbers the a + b is called additio of two umbers. a c ii) ad two fractios the b d a c ad bc b d bd e.g ) Subtractio a c ad bc b d bd e.g ) Multiplicatio a c a c b d b d 4 4 e.g ) Divisio a c a d b d b c a d e.g. b c where c d is called Multiplicative Iverse e.g. 4 5 (Now use rule of Multiplicatio) ) Order a ad b are two give umbers the possible relatio betwee these two are, i) a < b a is less tha b ii) a > b a is greater tha b iii) a = b a is equal to b Mathematics & Statistics / 6

29 e.g. a ad b are two studets i TMV. the a ad b must be admitted to ay of the 5 coerces ru by TMU. If a take admissio to B.C.A. b take admissio to M.C.A the by age a smaller tha b a ot bigger tha b ad a is ot same age as b. 6) Number lie All umbers we ca preset o a lie is called umber lie. - ve Numbers O + ve Numbers origi e.g. Represet the umbers o lie -,,, -5, Meas All positive umbers are o right had side of Number lie whereas all ve umber o Left had side... Divisibility Test Number a is called divisible by b whe we divide o. a by o. b ad remaider equal to zero If remai is ot equal to zero, we say that a is ot divisible by b Test of Divisibility ) Test of : A umber is said to be divisible by whe its uit place digit is oe of,, 4, 6, ad 8. T U e.g. ) 5 6 uit place digit = give umber is divisible by U ) uit place digit = 9 give umber is ot divisible by ) Test of : A umber is said to be divisible by, whe additio of digits of give umber is divisible by. e.g. ) = 5 5 = 5 Remaider = give umber is divisible by ) = = is ot Iteger =. - give o is ot divisible by ) Test of 4 : A umber is said to be divisible by 4, whe last two digits T ad U Fuctios / 7

30 is divisible by 4. e.g. ) 4 5 last two digits ot divisible by 4 give o. is ot divisible by 4 ) last two digits 4 4 is divisible by 4 give o. is divisible by 4 4) Test of 5 : A umber is said to be divisible by 5, whe the last digit is or 5. e.g. ) last digit = 5 give o. is divisible by 5 ) last digit = give o. is ot divisible by 5 5) Test of : A umber is said to be divisible by, whe last digit is. e.g. ) 5 last digit = give o. is ot divisible by ) 4 4 last digit = give o. is ot divisible by. Check your Progress State True/False ) is divisible by 5 ) is divisible by ) is divisible by 5 4) is ot divisible by 5) is divisible by, 5,, ad 4.,,, Natural Numbers.. PRELIMINARY CONCEPTS takes always value oe i ay case, cocept, example. so the umbers are called as costats. But whe value is ot costat e.g. Age of studet i F.Y. Probable year is after th std. meas above 7 years. But it may be 8, 9 also. So the age of studet is a variable ad deoted by small alphabets as, -, y, z, p, q, That meas the age of studet lies betwee 7 ad 9 ad above. These values are called itervals of variables. There are four types of itervals. ) Ope Ope iterval e.g. (,5) meas our variable takes values betwee ad 5. But ot ad 5. Mathematics & Statistics / 8

31 < x < ) Ope Closed < x 5 e.g. ( 5) meas variable takes values betwee ad 5 ad 5 also but ot. ) Closed Closed e.g. (, 4) meas Variable takes values ad 4 betwee all values ad 4. 4) Closed Ope (5, 7) meas Variable takes values betwee 5 ad 7 ad 5 but ot 7... Absolute Value x is a Variable. The x = x if x > o = x if x < o x ca be read as modx Imp. Results ) x y x y ) x y x y ) xy x y 4) x x if y y y.4 CORRESPONDENCE Defiitio If A ad B are two sets such that by some rule to a elemet a A takes oe or more elemets i B, the the rule is called a correspodece There are Four types of correspodece ) Oe Oe correspodece e.g. Marks i examiatio of studets. A = Set of Number of studets. B = Set of marks obtaied by studets. A = { a, b, c, d, e } B = { 9, 8, 6, 68, 7} 5 x< 7 Fuctios / 9

32 Ve diagram a b c d e A B ) May Oe Correspodece A = Number of subjects for F. Y. B = Number of subjects i F. Y. i TMU A = { Computer Network, VB Net, Math & stats office Automatio, Operatig system, C++. Accoutig} B = {a, b, c, d, e, f} C++ CN VB M&S OA OS ACC A Subjects of studet A take ) M & S ) OA ) C++ 4) OS 5) VB 5 papers (subject) for each studet. a b c d e f B ) May May A = {x, y, z} B = {a, b, c, d} x y z A B C d A B Mathematics & Statistics /

33 4) Oe May ) A = {,, } B = {a, b, c, d} A a b c d B ) A = Set of states B = Set of cities R : A B R : called as Rule of correspodece from oe set to aother set. Check your Progress.4. State the correspodece. Draw ve diagrams. i) A = {,, } B = {a, b, c, d} R : A B R = { (,a) (,b), (, c)} ii) A = {, 4, 6, 8} B = {p, q, r, s} R : A B R = {(,p), (,q), (4,r) (6,s), (8,r)} iii) A = {,, 5, 7} B = {p, q} R : A B R = {(,p), (,p), (5,q), (7,q)} iv) A = {, 4} B = {r, s} R = A B R = {(, r), (,s), (4,r), (4,s)}.5 FUNCTIONS Defiitio Let A ad B are two o empty sets A correspodece from the set A to set B is called fuctio if it is either oe-to-oe or may-to-oe. Every fuctio is correspodece but every correspodece is ot to a fuctio. Fuctio is a relatio R : A B (Relatio from A to B) for every x A there is uique y B such that x R y (x related to y). A fuctio is deoted by letters f, g, h. We write f : A - B g : C D h : C B if f : A B the xry x is related to y y = f(x) The set A is called as the domai of the fuctio The set B is called as the co-domai of the fuctio y = f(x) is called as Rule of correspodece y B is called as image of x uder. The set of images i B is called as the Rage of the fuctio. Rage of r = {y f(x) = y, where x A, y B} The Rage of a fuctio is subset of its co domai Fuctios /

34 Note If A B, the, ) Every elemet of set A is related to oe ad oly oe elemet of set B. ) More tha oe elemet ca be related to set A to oe elemet of set B ) Set B may cotai elemets which are ot related to ay elemet of set A. e.g. Let A = {,,,} B = {4, 5, 6, 7,} f : AB f = { (, 5), (,6), (,7)} the ve diagram A Set A is domai Set B is Co-domai Rage = {5, 6, 7} Images f() = 5- f() = 6 f() = 7 Rule f(x) = x + 4 = y y B Set of order pair = {(,5), (,6), (,7)} B Tabular form X y = f(x) Check your Progress. Fid the image of followig fuctios a) f(x) = x² - x + 4 fid f(), f(), f(-), f(-), f() b) f(x) = x² - 5 Fid f (-), f(-), f(), f(-), f() ) Oto fuctio ) Ito fuctio ) Oe-Oe fuctio 4) May Oe fuctio 5) Eve fuctio 6) Odd fuctio 7) Composite fuctio.6 TYPES OF FUNCTIONS ) Oto fuctio (subjective fuctio) Defiitio A fuctio F : A B is said to be a Oto fuctio if every elemet of set B is the image of some elemet of set A. R = B (Rage = codomai) symbolically, we write, oto Mathematics & Statistics /

35 f : A B e.g. A = {-, -, -,, } B = {, 4, 9} f : A B f(x) = x² A B ) Ito fuctio (subjective fuctio) Defiitio A fuctio f : A B is said to bea ito fuctio if there exits at least oe elemet i B, which is ot the image of ay elemet of A. RCB (Rage is proper subset of B) e.g A = {-, -,,, } B = {,,,, 4} f : A B f(x) = x² = y A B Rage = R = {,, 4} B = {,,,, 4} RCB Note : The fuctio is Oto or Ito which deped o Rage of that fuctio. ) Oe Oe fuctio (Ijective fuctio) Defiitio A fuctio f : A B is said to be Oe-to-oe fuctio if distict elemets of A have differet images i B uder f. e.g. A = set of studets. B = set of Roll Numbers R : A B Roll Number is fixed with studet. Amit s Roll No. is 8. meas No. 8 is Asie to Amit oly ad ot other, ay studet. oe-to-oe correspodece. oe-to oe fuctio. 4) May-oe fuctio Defiitio A fuctio f : A B is said to be may-to-oe fuctio if two or more elemets of A have the same image B i.e. there is at least elemet i B, which has more tha oe oe pre-image i A. 5) Eve fuctio Defiitio A fuctio f : A B is said to that meas f(x) does ot chage with x ad x replacemet. e.g. ) f : A B Fuctios /

36 f (x) = y = x² + Now put x = -x f(-x) = y = (-x)² + = x² + x² = (-x)² f(x) is eve fuctio. ) f(x) = x + 5 put x = -x f(-x) = (-x) + 5 = -x + 5 f(x) f(-x) Give fuctio is ot eve fuctio. 6) Odd fuctio Defiitio A fuctio f : A B is said to be odd fuctio if f(x) = -f(x) e.g. ) A fuctio f(x) = x³ + x f(-x) = (-x)³ + (-x) = (-x)³ -x = [x³ + x] = - f(x) Give fuctio is A Odd fuctio. ) f(x) = x² + 4x f(-x) = (-x)² + 4(-x) = x² - 4x = - f(x) Give fuctio is ot Odd fuctio. 7) Composite fuctio Defiitio A fuctio f : A B ad g : B C be two fuctios. x is ay elemet of A. The y = f(x) B. Sice B is the domai of fuctio g ad C is its co-domai, g(y) C so z = g(y) C This fuctio is called as composite fuctio of f ad g. Let it deoted by h. Thus gof = h : A C such that h(x) = g [f(x)] A B C f X Xy = f ( x ) Z = g ( y ) g h e.g. ) Let f : {,, 4, 5} {, 4, 5, 9} ad g : {4,, 5, 9} {7,, 5} fuctios defied as f() =, f() = 4, f (4) = = f(5) = 5 ad g () = g(4) =7, g(5) = g(9) = fid g of. Solutio ) Mathematics & Statistics / 4

37 give ) f(x) = g of () gof () gof (4) gof (5) = g[f()] = g() = 7 = g[f()] = g(4) = 7 = g[f(4)] = 9(5) = = g[f(5)] = 9(5) = A B C g f A h C h g of (x) = g[f(x)] g of (x) = g of () = g[f(x)] = 9[f()] = g() = 7 g of () = g[f()] = 9(4) = 8 gof() = g[f()] = g(5) = 9.6 Check your progress Fill i the blaks i) The fuctio is either oto or Ito which depeds o the give ii) a) Rage b) domai c) co-doma d) value A oe-oe fuctio is also called a) Ijective b) Bijective c) Oto d) Ito Fuctios / 5

38 iii) Oe-oe ad Oto fuctio is called a) Bijective b) Ijective c) Ito d) eve. iv) May-oe correspodece is called fuctio a) eve b) add c) Bijective d) May-oe. v) A may-oe fuctio ca be either a) ito or oto b) eve or odd c) eve or bijective d) ito or odd. vi) Nature of fuctio whether it is oe-to-oe or may-oe deped, upo of fuctio. a) Domai b) Co-doma c) Rage d) eve..7 GRAPH OF FUNCTION Defiitio A fuctio f : A B, x A ad y B the (x,y) be a elemet of f. We ca plot the poit (x,y) i a plae by choosig a suitable co-ordiate system. O plottig all such order pair, we get geometrical represetatio (curve) of fuctio f this is called graph of fuctio f. e.g. f(x) = x + X F(x) = y Y 5 4 Y=x X -.8 SUMMARY Fuctios meas a actio, the velatio betwee variable ad umber. Correspodeces are of four types. There are also types of fuctios. We ca draw proper ad et Ve diagram for each correspodeces. The et graph is there for each fuctio..9 CHECK YOUR PROGRESS - ANSWERS. ) False ) True ) True 4) True 5) True Mathematics & Statistics / 6

39 .5 ) a ) b c d p q r s A B A B oe-oe oe-may ) 4) 5 7 P q 4 A r s B A May-Oe B May -May.6 ) c) ) a) ) a) 4) d) 5) a) 6) a). QUESTIONS FOR SELF STUDY Problems For Practice. If f(x) = x² < x < fid f (-), f ( ), f(). Fid g of ad fog if f(x) = x + ad g(x) = x² - x + 4. If f(x) = x x = 6 < x < = x 6 x 4 Stat (a) Domai ad Rage (b) fid f(), f(-4), f() 4. f : R R, g : R R are defied as, f(x) = x² x = -x x > g(x) = x + x = x² x > fid g of (x), x R 5. Test whether followig fuctios are eve or odd ) f(x) = g - x² ) f(x) = x ) f(x) = x + 4) f(x) = x 5) f(x) = x x ± Fuctios / 7

40 6. Draw the graph of give fuctios ) f(x) = ) f(x) = x -x - R ) f(x) = x 4) f(x) = - if - x < - = - if - x < = if x < = if x fid images of x = - = = = 7. Fuctios f ad g are give by the followig i) f = { (, ), (, ), (, 4), (4, 5), (5, 6), (6, 7)} g = { (, 4), (, 6), (4, 8), (5, ), (6, ), (7, 4) } ii) If f = { (, ), (4, 5), (6, 7), (8, 9)} g = { (, ), (, 4), (, 6), (4, 8)} iii) If f = { (, ), (, ), (, 5), (4, 7) ), (5, 9)} g = { (, 5), (, 4), (, ), (4, ), (5, )} a) Express f ad g by formula b) Show that f ad g are oe-oe fuctios. c) fid fog ad gof. Problems for practice ) f(-) = -, f( ) = 4, f() does ot exits ) gof (x) = 6 x² - x + 9 fog (x) = x² + x + 8 ) Rage = [-,4] f() = 6, f(-4) does ot exist, f() = 4) gof (x) = 9 [f(x)] x g[f()] = x² = g[f(-) = x² = gof (x) = g[f(x)] = g [f()] = g(x²) = (x + )² = 4 gof (x) = 9[f(-)] = g() = x² = 5) ) eve ) eve ) eve 4) odd 5) odd. Mathematics & Statistics / 8

41 6) ) Y = f(x) = ) (, ). SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby. Statistical Aalysis: A Computer - Orieted Approach Itroductio to Mathematical Statistics by S. P. Aze & A. A.Afifi Fuctios / 9

42 NOTES Mathematics & Statistics / 4

43 CHAPTER Sequeces, Progressios ad Series. Objectives. Itroductio. Sequece.. Summatio of terms of a sequece. Arithmetic Progressio.. The th term of A.P. (T ).. Sum of first terms of A.P. (S ).4 Geometric Progressio..4. The th term of G.P. (T ).4. Sum of first terms of G.P. (S ).5 Harmoic progressio (H.P.).6 The three Meas.6. Properties of meas.7 Series.7. Stadard series.7. Ifiite Geometric series.8 Summary.9 Check your Progress - Aswers. Questios for Self Study. Suggested Readigs. OBJECTIVES After studyig this chapter, you ca explai ad use various types of sequeces, series give below : Sum of terms of a sequece. Three types of sequeces. Arithmetic progressio. Geometric progressio. Harmoic progressio. The three meas : A.M., G.M. ad H.M. Summatio of series of terms. Summatio of certai ifiite series i G.P.. INTRODUCTION I computer applicatios may cocepts ca be expressed as a ordered umbers usig,. We have already see oe such case of expressig a set or a fuctio formed by various combiatios of,. Briefly the ordered set of umbers forms a sequece. We shall the -cosider three types of sequeces ad summatio of terms of sequeces formig a series.. SEQUENCE We kow that the system of atural umbers is,,,..., +,... This is a collectio of umbers satisfyig the followig properties. (i) It is ordered i.e. each umber of the collectio has a defiite positio. (ii) There exists defiite law, accordig to which every umber ca be writte dow. (iii) Every umber is followed by ext oe. Ay other collectio of umbers which satisfies the first two of the above Sequeces, Progressios ad Series / 4

44 properties is called a sequece. The sequece is further called ifiite sequece, if the third property is also satisfied, otherwise it is called a fiite sequece. Defiitio : A ordered set of umbers formed accordig to a well defied law is called a sequece. I terms fuctio, it is a fuctio f : N R If N, the f() is called th term of the sequece. By givig differet values to, we get the correspodig term of a sequece. e.g. (i),,,...,,... Here f() =, i.e. square of a atural umber. Whe =, f() = is the rd term of a sequece. (ii),,,..., 4.. Here f() = (iii), 5, 8,, 4,... Here f() = or f() = + ( ) or it is a sequece i which each term is obtaied by addig to the previous term. (vi) (iv), 4, 8, 6,,... Here each term is a power of. f() = is the th term. (v).,.,.4... Here f() = ( + ) is the th term. All these are illustratios of ifiite sequeces. Now we shall cosider certai fiite sequeces. The sequece,,,,,,,,,, is a fiite sequece with repeated terms. The digit, for, example occurs as the d, rd, 5 th, 6 th ad 8 th elemets of the sequece. The correspodig set is simply {, }, i which the order of 's ad 's is ot specified. (vii) A ordiary word i Eglish, such as 'Physics' ca be viewed as a fiite sequece. p, h, y, s, i, c, s composed of letters from the ordiary alphabets. If we omit commas, we get the word physics. Such represetatio is referred to as strig. I computer sciece a sequece is sometimes called a liear array or list. A array may be viewed as a "sequece of positios" which we represet below as boxes. The positio form a fiite or ifiite list, depedig o the desired size of array. Elemets from some set may be assiged to the positios of the array. The elemet assiged to positio will be deoted by S() [correspodig to f() i the defiitio] ad the sequece S(l), S(), S(), S(4), will be called the sequece of values of the array S... Summatio of terms of a sequece: Aother problem related with sequeces is to fid out the sum of first terms. It is tedious to add terms oe after aother i case of large umber of terms. It is therefore ecessary to fid a law givig sum of the first terms. Mathematics & Statistics / 4

45 Let T ad S deote the th term ad sum of first terms of a sequece respectively. Thus S = T +T T = T read as "sigma T r, r varyig from to " With this otatio, the sum of first ( ) terms is S = T r = T + T + + T r Hece S S, = T, true for. We shall illustrate it by simple examples. r Example () : If T = , fid T 4, T 9 ad T Solutio : We have T = Example (): If S = Solutio: puttig = 4, 9 ad respectively, T 4 = (4 ) + 4 (4 ) + 7 = 7 T 9 = ( 9 ) + 4 ( 9 ) + 7 = 86 T = ( ) + 4 () + 7 = 566. Fid T 5 ad T 7. We have S S, = T puttig = 5, we get S 5 S 4 = T 5.T 5 = = 5 = 5. Similarly T 7 = S 7 S 6 7(7 ) 6(6 ) = = 8 = 7. Example (): Fid T 5 give that S = ( ). Solutio : We have S S, = T puttig = 5, we get, T 5 = S 5 S 4 = = = 4 = ( 4 ) = 5 Sequeces, Progressios ad Series / 4

46 Check your progress. () The array represets a ifiite sequece Fill i the blak. () Choose correct figure from the bracket to fill i the blak. (i) T = +,T =.. (4, 5, 6, ) (ii) S = 6, T =...(4, 6, 8, ) (iii) S =, T =... (,,, + ). ARITHMETIC PROGRESSION Amogst the three types of sequeces which are very commo, the arithmetic progressio is oe of them. Cosider a sequece, 5, 8,, 4,... Here we see that 5 = 8 5 = 8 =... = i.e. the differece betwee a term ad its precedig term is costat. Such a sequece is called arithmetic progressio; abbreviated as A.P. Defiitio : A arithmetic progressio (A.P.) is a sequece i which the differece betwee ay term ad the immediately precedig term is costat. This costat differece is called the commo differece of the arithmetic progressio ad is usually deoted by d. I the above example the commo differece is ad first term is. The geeral form of a arithmetic progressio is where a, a + d, a + d, a + d,... a = first term, ad d = commo differece... The th term of A.P. (T ) Cosider a geeral A.P. a, a + d, a + d, a + d,... We observe that T = a = a + ()d =a + (l l)d T = a + d = a + ( l)d T = a + d = a + ( ) d etc. We ote that every term is obtaied by addig to 'a' certai multiple of d; ad this multiple is exactly oe less tha that of the suffix of T. T = a + ( l)d. Example () : Fid T for the followig A.P., 5, 9,, 7 Solutio : Here T = a = ad commo differece d = 4 T = a + ( l)d = l+( l)4 = 4. Mathematics & Statistics / 44

47 Example (): Fid the umber of terms i the A.P., 4, 7,..., 8. Solutio: Here a =, d = 7 4 = Let 8 be the th term of A.P. 8 = + ( ) i.e. 8 = ( l) = 7 ad = 8. Thus there 8 terms i the give A.P. Example () : If the th term of a sequece is +, show that it is a A.P., what is its first term ad the commo differece? Solutio : Here T = + Replacig by ( ) we get, T _ = ( l) + = l Cosider, T T l = + ( ) = This is costat (idepedet of ) Hece it is th term of A.P. with d =. st term = T = () + = 5 ad the commo differece is ( = T T _ )... Sum of the first terms of A.P. (S ) Let S deote the sum of st terms of the geeral A.P. with T = a ad T = a + ( l)d Let T = a + ( )d= l The we have T = l d, T,= l d Now S = T +T + T T + T +T Reversig the order of terms i summatio, Addig vertically, we get = a + (a+d) + (a+d) (l d) + (l d) l S = l + (l d) + (l d) (a+d) + (a+d)+a S = (a+l) + (a+l) (a+l) + (a+l) to terms = ( a + l) S = (a + l) Substitutig the value of, we get, S = {a + ( l)d} We shall illustrate the use of these formula i certai simple examples. Example (4) : Fid T ad S for the followig A.P. 9, 5,, 7,... Solutio : Here a = 9, d = 5 ( 9 ) = 9 5 = 4 T = a + ( ) d = 9 + ( ) (4 ) = 4. Sequeces, Progressios ad Series / 45

48 ad S = { a + ( ) d } = { 58 + ( l)4 = ( l). Example (5): Fid the sum of the first odd atural umber Solutio : The first odd atural umbers are,, 5,... T = th odd atural umber = l+( l) (a =, d = ) = = l say S = (a + l) = =. ( + l) Example (6) : If the sum of the first terms of a sequece is + 4, show that it is a A.P. Fid the first term ad the commo differece. Solutio : Here s = + 4 Replace by ( ) to obtai S. S _ = ( ) + 4 ( ) = = T = S S = + 4 ( l) = 6 + l. Replace by ( ) to obtai T _ T = 6( ) + = 6 5. Cosider, T T _ = 6+ (6 5 ) = 6, which is costat. Hece S = +4 is sum of first terms of a A.P. with d = 6. The first term = T = 6() + =7 or S = + 4 = 7 Check your progress. () State whether true or false. If false write correct aswer. (i) Numbers a d, a, a + d are i A.P. (ii) For a A.P., T T _ is ot costat. (iii) For a A.P., T 8 = 6 the S 5 = 54. (iv) I a A.P., s = the a = T = 4. (v) I a A.P., S = the d = 6. (vi) The sum of first eve atural umbers is ( + ). (vii) For a A.P. a =, T 7 =, the commo differece is 7. (viii) For a A.P. S = 86, the T 6 = 6. (ix) For a sequece havig terms, S = Mathematics & Statistics / 46 the the sequece is a A.P.

49 .4 GEOMETRIC PROGRESSION Cosider a sequece, 9, 7, 8,... We observe that the ratio of ay term to its precedig term is = ad is costat. Such a sequece i which ratio of 9 7 a term to its precedig term is costat is called geometric progressio, abbreviated as G.P. Defiitio: A geometric progressio (G.P.) is a sequece i which every term bears a costat ratio to the oe immediately precedig it. This costat ratio is called the commo ratio of the G.P. ad is usually deoted by r. I the above example the commo ratio is. The first term is also here. The geeral form of a geometric progressio is a, ar, ar,... where a is the first term, ad r is the commo ratio..4. The th term of G.P. (T ) Cosider a G.P. a, ar, ar... With the usual otatio, T = a = ar = ar T = a r = a r T = ar = ar We ote that each term is a product of two factors. The first factor is a, ad is commo to all the terms. The secod factor is a certai power of r. This is exactly oe less tha the correspodig suffix of T. T = ar Example () : Fid T for the followig G.P.,, Solutio : Here a =, r = / = / = / = T = a r = =,,... 4 Example (): Give T =, T 7 = of a certai G.P. Fid T 8. Solutio : We have T = a r = ad T 7 = a r 6 = where a is the first term ad r is the commo ratio of G.P. O divisio, we get, a r a r The a r = gives a = 6 i.e. r 4 = 6 r = r 4 = 5.4. Sum of first terms of a G.P. (S ) T 8 = a r 7 = 5 ( 7 ) = 64. Let a be the first term ad r be the commo ratio of a geeral G.P. The T = ar ad Sequeces, Progressios ad Series / 47

50 S = a + ar + ar ar + ar Multiply by r. rs = ar + ar + + ar + ar + ar. (II) Subtractig (II) from (I), we get S rs = a ar Divide by ( r). i.e. S ( r) = a ( r ) a S r = r ar =, if r <, if r > r However if r =, G.P. becomes a, a,... Here S = a + a +... to terms = a We shall illustrate the use of these formula by meas of some simple examples. Example () : Fid T ad S for the followig G.P., 9 7,,, Solutio: Here a =, r = 9 4 T = a r = a ad S r = r 5 Example (4): For a G.P. a = 5, r =, S = 65, fid. a Solutio : Here a = 5, r =, S r = r 5 65 = 7 = 8 = i.e. 7 = = 7 Check you progress.4 () Fill i the blaks by choosig appropriate umber give i the bracket (i) I a G.P. with a =, r =, T =... (8,,,9) b (ii) a, b,... are i G.P. the the commo ratio is... a a b,b,a, b a (iii) If for a sequece, S = (4 ), the the sequece is... (A.P., G.P., oe of these) (iv) I a G.P. a = ad T6 = 6 the the commo ratio r =... (,, 4, 6) Mathematics & Statistics / (I)

51 (v), 9 7,, are i (A.P., G.P., H.P., oe of them) () I a G.P. T = T 7 = fid a ad r. () I a G.P., a = 5, r = 5 fid T 5. (4) State whether true or false ad if false write the correct statemet, (i) I a G.P. S =, the commo ratio is. (ii) I a sequece, 6, 8,... the 4 th term is 6. (iii) I a G.P.,,, the th term is. a a a a (iv) I a G.P. T = ad T5 = the T4 = ± HARMONIC PROGRESSION (H.P.) Cosider a sequece,,,,...(a) 4 The reciprocal of the terms of the sequece, form aother sequece viz.,,, 4,..(B) The terms of the sequece (B) are i A.P. Such a sequece (A), i which the reciprocals of the terms form A.P. is kow as Harmoic progressio, ad abbreviated as H.P. Defiitio : The terms a, a,a,..., a,... are said to be i harmoic progressio (H.P.) if.. a, a, a,... a be i arithmetic progressio. To fid th term of H.P., we have to fid th term of the correspodig A.P. Example: Fid T ad hece T 5 ad T 7 i the followig harmoic progressio.,,,... 7 Solutio: The correspodig A.P. obtaied by takig reciprocal of the terms i the give sequece is 7,,,... Here a = 7, d= Hece T of A.P. is a + ( ) d The th term of give H.P. is, = 7+( ) T = Hece puttig = 5 ad 7, we get T 5 = T 7 = There is o coveiet formula that ca be developed to obtai sum of first terms of H.P. ad Sequeces, Progressios ad Series / 49

52 Check you progress:.5 Aswer the followig. () Give A.P. as,, 5, 7, 9,... Write dow the correspodig H.P. () I a harmoic progressio first two terms are ad, a b differece of correspodig A.P. Mathematics & Statistics / 5 fid the commo.6 THE THREE MEANS If x, y, z, are cosecutive terms of a sequece the y is called mea of x ad z. We shall? cosider the three meas correspodig to three progressios that we have studied so far; ad! also study some of its properties. Defiitio : If a, A, b are cosecutive terms of a A.P. the A is called arithmetic mea betwee ad a ad b ad is abbreviated as A.M. Sice a, A, b are i A.P., the commo differece of A.P. is = A a = b A i.e. A = a + b i.e. A = a b Thus the A.M. betwee two umbers a ad b is e.g. (i) The A.M. of ad 5 is,,,, = (ii) The A.M. of ad is. 4 a b =A Defiitio : If a, G, b are all positive ad are three cosecutive terms of G.P. the G is called geometric mea betwee a ad b ad is abbreviated as G,M. Sice a, G, b ( a >, G >, b > ) are i G.P. the commo ratio = i.e. G = ab i.e. G = Thus the G.M. betwee two positive umbers a ad b is G = e.g. (iii) The G.M. of 5 ad 5 is 5 5 = 5 ab a (iv) The G.M. of ad a r is ar r a r = a. ab G a Defiitio : If a, H, b are cosecutive terms of H.P. the H is called the harmoic mea betwee a ad b. Sice a, H, b are i H.P., the reciprocals of them viz a, H ad b are i A.P. The commo differece of A.P. is = i.e. H = H a a b a b i.e. = H ab i.e H = ab a b b H b G

53 ab Thus the H.M. betwee ad b is H = a b e.g. (v) The H.M. of ad is (vi) The H.M. of 4 ad 7 is.6. Properties of meas Let a, b be the give uequal positive real umbers. We have A.M. is A = G.M. is G = a b ab ab H.M.is H = a b We have the followig properties. () A, G, H are i G.P. or AH = G a b ab a b Proof :Cosider, AH = ab ab G () A > G > H. (a >, b > ) a b Proof: Cosider A G = ab Sice a, b are positive. :. b = a b a b = a, b are real. a a b AG A > G...(I) or A / G > A G By property, () = G H ab. ad by (I), G A H G G > H... (II) It ow follows from (I) ad (II) that A > G > H. This- gives a very coveiet way to assume, 4, 5 umbers i the respective progressios. We prepare the followig table. progressio os. 4 os. 5 os. A.P. a d, a, a + d a d, a d, a + d, a + d a d, a d, a, a + d, a + d G.P. a, a, a/r, a/r, ar, ar a/r, a/r, a, ar, ar r Sequeces, Progressios ad Series / 5

54 We shall illustrate the use of the meas i the followig examples. Example (): Fid three umbers i A.P. such that their sum is 5 ad their product is 5. Solutio : Let the three umbers i A.P. be a d, a, a + d. Their sum is 5. ( ad) +a + (a + d) = 5 i.e. a = 5 a = 5 Hece umbers are 5 d, 5, 5 + d. Their product is 5. (5 d)(5)(5 + d) = 5 5 d = d = ± Hece the umbers are 5, 5, 5 + or ( d = ), 5, 7 or 7, 5, are the umbers. 5 +, 5, 5 ( d = ) Example () : Fid four umbers i G.P. such that their product is ad the sum of two middle is 5. a a ar r r Solutio : Let, ar,ar, Their product is. a a r r Hece the umbers are arar r be four umbers i G.P. = a 4 = a = r,,r, r The sum of the two middle umbers is 5. 5 r = r i.e. +r 5r = i.e. r 4r r + = i.e. (r l)(r ) = Either r = or r =. Whe r =, umber are,,, 8 8 or whe r =, umber are 8,,, Example () : If 6 is G.M. ad is the H.M. betwee two umbers, fid them. Solutio : Let a, b be the two umbers. G.M. is 6. ab = 6 Squarig we get ab = 6...(I) Mathematics & Statistics / 5

55 7 Their H.M. is ab a b = i e = a b i.e. a + b = cosider, (a b) = (a + b) 4ab = = 5 a b = 5 ad a + b = Addig, we get a = 8 a = 9 ad b = 4 Thus 9 ad 4 are the required umbers. Example (4) : If A.M. ad G.M. betwee two umbers be ad respectively. Fid their H.M. Solutio : Let a, b be the two umbers. 7 6 Their A.M. is 7 = 9 9 ad G.M. is = a b = i.e a + b = The H.M is give by H = i.e. H.M = = 6 ad 9 9 ab ad ab = ab a b 9 5 H.M. betwee the umbers is 5 Check your progress -.6 Aswer the followig : 9 () Give two umbers ad 8, fid A.M., G.M. ad H.M. () a ad b are two umbers such that their A.M. is ad H.M. is, show that a = ad b =. () If a is A.M. betwee b ad c, b is G.M. betwee c ad a, show that H.M. betwee a ad b is c. (4) Fid five umbers i G.P. such that their product is ad the product of the last two umbers is 8. (5) If 4, H, 9 are i H.P. fid H. 9 Sequeces, Progressios ad Series / 5

56 .7 SERIES We have see that f : N R is a sequece; with f() as the th term of a sequece. Now f () is called a series. If it cotais sum of a fiite umber of terms, the it is a fiite series. We shall cosider maily a fiite series..7. Stadard series () We assume certai stadard results without proof. () () (4) r = = r = l r = l = (ar + br + cr + d) = a 6 4 Mathematics & Statistics / 54. r +b. r + c We shall cosider examples i which these results are used. Example (): Evaluate r (r +) (r + ) Solutio :We have r (r + ) (r + ) = r(r + 4r + ) = r + 4r + r Hece the give sum = = = = = = = (r ) + 4 r + (d) (r ) + (r) 4 4 [(+)+8(+)+8] [ +9+6] [ +6++6] [(+)(+)] ( )( )( ) Example () : Fid T r ad sum to terms of the followig Solutio : The st factors i each term are, 5, 8,... These are i A.P. with a =, d = T r = + ( r ) = r The d factors are 5, 8,,... These are also i A.P. with a = 5, d = T r = 5 + ( r ) ( ) = r + T r = T. T =(r l)(r + ) r r 6

57 Now S = = 9r + r = 9 = 9 T r = r + (9r + r ) r 6 = = [9(+l)( + l) + 9(+l) ] 6 [9( + + ) + 9 ] 6 = [ ] 6 = ( l). Example (): sum to terms of Solutio : Here we shall express each term as a differece of two terms, oe of the which is a geometric series. Let S = to terms = S S say Now S = to terms. These terms are i G.P. with r = a =, =. = () + ( )+( ) +... to terms = [ to terms] [++... to terms] ( ) S = ( ) 9 ad S = to terms = S = S S = 9. Example (4) : Fid the sum of the first terms of the followig arithmetico - geometric sequece.,, 4, 4 8, 5 6, 6,... Solutio : The st factors are,,, 4,... T = r The d factors are,,,, 4,... These factors are i G.P. with a =, r = r T r = l() r = r T r = T r Hece S = T r T r = r r = (A) The commo ratio of G.P. is. Hece multiply S by Subtract (A) from (B) S = ( l) + S S = ( l )+ = (l ) Sequeces, Progressios ad Series / (B)

58 S = S where S = are i G.P. with a =, r =, =. S = = Hece S = ( l) = ( l) + l. Example (5): Express the followig recurrig decimals as ratioal umbers. 45 Solutio : We have. 45 = = 5 45 The terms after the first term form a G.P. whose first term is commo ratio r = = a ad the Now r = < ad whe becomes very large, r almost becomes zero. a 45/ Sum of these terms i G.P is r / = Hece the give umber. 45 =.7. Ifiite Geometric Series Mathematics & Statistics / Cosider a geometric progressio with T =. The terms are,, We observe that as becomes larger ad larger the terms of a G.P. go o becomig smaller ad smaller. If we take to be very large, the correspodig term approaches zero. We say a S r = approaches zero. r This happes oly whe r <. becomes / For values of r such that r >, such a sum does ot exist. Defiitio : If for a G.P., S approaches a certai umber S as becomes idefiitely large, the umber S is called sum to ifiity of the G.P. Let a be the first term ad r (< ) be the commo ratio. a S r = r a S = r

59 Thus a + ar + ar ar a +... = r We shall explai this cocept by the followig example. Example (6): Fid S ad sum to ifiity of the followig G.P ,,,,, Solutio : Here a = 5 ad r = 5 5 a S r 5 5 = r 4 4 Sice r = = <, the sum to ifiity of the G. P. exists ad a 5 / 5 S=. r 4 Example (7): The first term of a G.P. is ad the sum to ifiity is 6. Fid the commo ratio. Solutio : Here a =, S = 6 We have to fid r. We kow that S = 6 = r = r = Thus the commo ratio is. Check your progress.7 a r r 6 () Determie whether the sum to ifiity of the followig G.P.'s exits. Determie it, whe it ; exists (i),, 4, 8. 6,... (ii) 4,,,,, (iii), ,,,... (iv),,,, (v),,,, () Give that to terms = S = ( ), fid the followig sums i terms of S. 9 Sequeces, Progressios ad Series / 57

60 (i) to terms (ii) to terms () Show that the recurrig decimal. 56 is. 99 (4) Prove that the sum to terms = 6 ( ) (5) If the sum of ifiity of the sequece, 5r, 7r 8,... is 4, fid r. 9.8 SUMMARY Mathematics meas umbers ad sequece meas arragemet of proper umber i proper way. E.g.,,, 4, 5 give umbers are icreasig ad icreased with oe uit. So it is sequece. Series is cocept of additio of sequece terms.. (), 4, 6, 8,....9 CHECK YOUR PROGRESS - ANSWERS () (i) 4 (ii) 4 (iii) ( ). () (i) True (ii) False; For a A.P., T T is costat. 5 (iii) True, Hit : S 5 = ( T8 ) (iv) False; I a A.P., S =, the a = T = (v) True, Hit : S = a = T =, S = a + d = 8 (vi) True (vii) False; For a A.P. a =, T 7 =, the commo differece is. (viii) True, Hit : S = (T6 ) (ix) True..4 () (i) 8 (ii) a b (iii) G.P. (iv) (v) G.P. () a = 5, r = () 4 5 (4) (i) True (ii) False; I a sequece, 6, 8,...the 4 th term is 54 (iii) False; I a G.P.,, True..5 (),,,,,... () b a (), 48 6, (4),,,6, 8 9 Mathematics & Statistics / 58,,,.. the th term is a (iv) a a a

61 (5) 7.7 () (i) No (ii) yes, S = 6 (iii) yes, S = (iv) No (v) S =, yes 5 5 () (i) 9 5 S (ii) 9 S (5) a dr. Hit: 4 4 r 9 r 8 with a =, d =. Self Study Problems.9 QUESTIONS FOR SELF - STUDY () Fid the th ad th term of the Sequece, 4,8, Also fid the sum of First terms. () I a G. P. the third term is ad fifth term is fid th term of G.P. 8 6 () Fid three umbers i a G. P. Such that their sum is 7/ ad sum of their squares is 4 (4) The Sum of First terms of a Sequece is (4 ) show that it is a G. P. Fid its Commo Ratio. (5) How may terms of the A. P., 7,... are eeded to yield the sum 75?. SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Mathematics ad Statistics by M. L. Vaidya ad M. K. Kelkar Sequeces, Progressios ad Series / 59

62 NOTES Mathematics & Statistics / 6

63 CHAPTER 4 Permutatios ad Combiatios 4. Objectives 4. Itroductio 4. Multiplicatio Priciple 4. Factorial Notatio 4.4 Permutatio 4.4. Permutatios of thigs ot all differet 4.5 Combiatio 4.6 Summary 4.7 Check your Progress - Aswers 4.8 Questios for Self - Study 4.9 Suggested Readigs 4. OBJECTIVES After studyig this chapter you will be able to use ad explai the followig ideas very freely ad cofidetly. Permutatios Permutatios of thigs ot all alike Combiatios Complemetary combiatios 4. INTRODUCTION We shall start with a simple but geeral result kow as multiplicatio priciple i combiatorics or fudametal priciple i old laguage of mathematics. This has umerous applicatios i this chapter ad else where. It will followed by permutatios ad combiatios. 4. MULTIPLICATION PRINCIPLE Let us suppose that two tasks T ad T, are to be performed i a sequece. A task T ca be performed i ways ad for each of these ways a task T ca be performed i 4 ways. We have to determie a sequece of performig tasks T T. We shall exhibit this by meas of followig tree diagram. Permutatios ad Combiatios / 6

64 Each task T ca be performed i ways ad after that for each of these tasks (T ), a task T ca be performed i 4 ways. Thus there are 4 = ways of performig tasks T, T i a sequece. If a task T ca be performed i m ways ad after performig the task T i ay of these m ways a secod task T ca be performed i differet ways, the possible ways, of performig tasks T ad T i a sequece is m. This ca obviously be geeralised as follows : If a third task T ca be performed i p differet ways, the possible ways of performig tasks T, T, T i a sequece is mp ad so o. We shall ow illustrate this priciple by the followig examples. Example () : A label idetifier, for a computer program cosists of oe letter followed by three digits. How may distict lable idetifiers are possible (i) if o digit is repeated, (ii) if repeititio of digit is allowed? Solutio : We cosider a array cosistig of four empty boxes. (i) (ii) Whe oe of the digit is repeated refer to array (A). The successive boxes ca be filled i 6,, 9, 8 ways. Hece by the exteded multiplicatio priciple, possible label idetifiers are = 87. Whe ay of the digit is repeated with referece to array (B), the successive boxes ca be filled i 6,,, ways. Hece by the exteded multiplicatio priciple, possible label idetifiers are 6 = 6. Mathematics & Statistics / 6

65 Example () : Show that the umber of subsets of a set cotaiig elemets is. Solutio : We use the cocept of characteristic fuctio of the set A havig elemets. F A cosists of a array of boxes, ad each box ca be filled i by ways viz, s or l's. Thus by the exteded priciple of multiplicatio, there are A.... factors = ways of fillig the array, ad therefore subsets of Example () : Let set A cotai elemets ad r. The umber of sequeces of legth r (allowig repetitios) that ca be formed from elemets of A is r. Solutio : A sequece of legth r ca be formed by fillig r boxes i a array. Let T be the task of fillig box. We ca choose ay of elemets of A to fill it. Sice repetitio is allowed the task T of fillig secod box ca be doe i ways. This is true for all the boxes. T r, the task of fillig r th box ca also be doe i ways. By the exteded multiplicatio priciple, the umber of sequeces that ca be formed is... to r factors = r. Example (4) : I how may ways ca the first ad secod prize i mathematics ad first ad secod prize i physics be awarded i a class of 5 studets. Solutio : The first prize i Mathematics ca be awarded i 5 ways ad havig doe this i ay oe way, a d prize ca be awarded i 4 ways. Thus task of awardig prizes is mathematics ca be doe i 5 4 = 6 ways. As awardig prize i Physics is irrespective of whether studet has obtaied a mathematics prize or ot, the task of awardig prizes i Physics ca be doe i 5 4 = 6 ways. By the multiplicatio priciple both tasks ca be performed i 6 6 = 6 ways. Check your progress 4. () A coi is tossed four times ad the result of each toss is recorded. How may differet sequeces of heads ad tails are possible? () A six faced die is tossed four times ad the umbers show are arraged i a sequece. How may differet sequeces are there? () I how may ways ca I write letters to three out of 9 frieds, if I have exactly oe post-card, oe ilad letter ad oe evelope? (4) Four persos eter a first class railway compartmet i which there are 6 seats. I how may ways ca they take their seats? 4. FACTORIAL NOTATION Let be ay positive iteger or ay atural umber. The product of first atural umbers ( l)( )..... is deoted by! or _ ad is read as "factorial " Thus! = ( ) ( )..... We defie o! = ad! =! = = Permutatios ad Combiatios / 6

66 ! = = 6 4! = 4 = 4 5! = 5 4 = ad so o. We ca also write! = () or! = ( ) ( ). 4.4 PERMUTATION Cosider a set A cotaiig elemets. If we select r elemets out of them ad arrage i a array each such arragemet is called permutatio. We ca ow formulate the problem i terms of subsets of A, each subset havig r distict elemets. Cosider a array of r boxes. The task T of fillig Box ca be doe by choosig "ay of '' elemets of A. Hece task T ca be doe i ways. Whe this has bee doe task T of fillig Box ca be doe i ( ) ways (elemet is ot to be repeated). Cotiuig this way the task T r, of fillig a Box (r ) ca be doe i (.r) = r + ways. The last task T r of fillig rth box ca be doe i r + l = r + l ways. By the exteded multiplicatio priciple, the tasks T, T,..., T r ca be doe i (l)()... ( r + ) ways. This umber is deoted by P r or of thigs" Pr Mathematics & Statistics / 64 ad is ofte called as "r permutatios Thus the total umber of subsets cotaiig r elemets out of a set A cotaiig elemets is P r = (l)()... (r+) I terms of factorial otatio, we get P r =... r r r r r... = ( ) ( )... /( r)( r l)... =!. r!... Note : Here the order of selectig elemets is importat. i.e. st elemet is selected, d elemet is selected ad so o. Alteratively, selectig r elemets out of give elemets ad arragig them i a row gives rise to P r permutatios. We shall illustrate it by meas of the followig examples. Example () : Fid the umber of differet arragemets that ca be made by usig all the letters of the word absurd. Solutio : The word absurd cotais 6 differet letters. The correspodig arrays has six boxes. Hece umber of possible arragemets of letters is 6P 6 = 6! = 7. Example () : Fid r, if rp 4 = 44. Solutio : We kow that rp 4 = r(r l)(r )(r ). We have to factories 44 ad arrage the factors i a array havig 4 boxes. 44 = 4 () = 4 7 4

67 = 4 7 = 4 r(r l)(r )(r ) = 4 r = 4. Example () : Show that the umber of permutatios of differet thigs take all at a time such that two particular thigs are ot together is ( ) l. Solutio : The umber of permutatios of all thigs without ay restrictio is P =! Whe two particular thigs say T, T are tied so that they are always together ca be arraged i ( ) P ( ) =( )! ways. But i each such arragemet T, T ca iterchage their positios. Hece by multiplicatio priciple, the total umber of arragemets, whe two particular thigs are always together are ( )! Thus the umber of permutatios of thigs take all at a time whe two particular thigs are ot together is! ( )! = ( )! ( )! = ( )! [ ] = ( ) l Example (4): A umber of five differet digits is to be formed with the help of digits,,,4,5,6,7 i all possible ways, (a) How may such umbers ca be formed? (b) How may of these are greater tha 4? Solutio : We have seve digits,,,4,5,6,7. Out of these just five digits are to be chose to form five digit umber. (a) We have to fill five boxes i the array. This ca be doe as follows. By usig exteded multiplicatio priciple the umber of digits so formed is 7P 5 = = 5. (b) A umber greater tha 4 ca be formed i two ways. (i) Box is filled by a umber (ii) Box is filled by a umber by choosig ay of 4 digits viz. 4,5,6,7. Case (i) : Box ca be filled by i oly oe way. The Box ca be filled by ay of 4 digits viz. 4,5,6,7. Hece Box ca be filled i 4 ways. The remaiig three boxes ca be filled by choosig ay of remaiig 5 digits i 5 P ways. Hece by exteded multiplicatio priciple, the umber of digits that ca be formed is 4 5 P = = 4. Case (ii) : Box ca be filled by ay of 4 digits viz. 4,5,6,7 i 4 ways. After this remaiig 4 boxes ca be filled by ay of the remaiig six digits i 6P 4.=6 5 4 = 6 ways. By exteded multiplicatio priciple, the umber of digits thus formed are 4 6 P 4 = 4 6 = 44. As the two cases are mutually exclusive the total umber of umbers greater tha 4 is their sum. Hece = 68 umbers ca be formed., Permutatios ad Combiatios / 65

68 Example (5) : I how may ways ca letters of the word MOBILE be arraged? I may of these, the cosoats occupy the eve places? Solutio : There are 6 letters i the word mobile. These ca be arraged i 6P 6 = 6! = 7 ways. Amogst these letters M, B, L are cosoats, ad O, E, I are vowels. The eve places are Box, Box 4 ad Box 6 i a array of 6 boxes. These three boxes ca be filled by the three cosoat i P =! = 6 ways. Also the remaiig three boxes ca be filled by the three vowels i P =! = 6 ways. By usig the exteded multiplicatio priciple, the umber of arragemets havig cosoats at eve places is 6 6 = 6 ways. Example (6) : Six boys ad seve girls are to be seated for a photograph i a row. Fid the umber of ways i which they ca be seated, if o two girls sit together. Solutio : Six boys ca be seated i a row i 6 P 6 = 6! = 7 ways. As o two girls are to be together, there are seve places for the girls to be seated as show i the above array. They ca occupy the seats i 7 P 7 = 54 ways. By the exteded multiplicatio priciple, the required umber of seatig arragemets is 7 54 = 688. Check your progress () Compute P () I how may ways ca twelve persos of which 6 are me ad six are wome be seated i a row if (i) ay perso may sit ext to ay other perso (ii) me ad wome must occupy alterate seats? () Fid the umber of differet permutatios of the letters i the word "group"? (4) I how may ways ca six people be seated i a circle? (5) Fid if P = ( lp ) + (7P ) (6) Fid r if (56 P r + 6 ) : (54P r + ) = 8 : (7) Eight papers are to be set at a examiatio, two of which are o computer. I how may orders, ca the papers be arraged, so that the two computer papers are (i) cosecutive (ii) ot cosecutive? (8) How may umbers lyig betwee ad 6 ca be formed by usig the digits,,4,5,7,9? 4.4. Permutatios of thigs ot all differet We shall cosider a case whe all the thigs are ot all the same, some, of them alike, aother of them are of same, of them are similar. The umber of district permutatios i this case are give by,!!!! We omit the proof ad see certai examples. Mathematics & Statistics / 66

69 Example (7) : Fid the total umber of distict permutatios of the letters of the word costitutio. Solutio : The word costitutio cotais letters of which T occurs times, each of N, O, I occurs times ad the remaiig C, S, ad U occurs oce oly. Hece the total umber of distict permutatios of these letters is =! (!) (!) (!) (!)! = = 6! 4 Example (8) : I how may ways ca 4 red, yellow ad gree discs be arraged i arrow if the discs of the same colour are idistiguishable? Solutio : The total umber of give discs are 4 (red) + (yellow) + (gree) = 9 discs. the total umber of arragemets 9! 4!!! = 6 Example (9): Fid the umber of arragemets of the letters of the word INDEPENDENCE. I how may of these arragemets, (i) (ii) Do the words start with P Do all the vowels always occur together (iii) Go the vowels ever occur together (iv) Do the words begi with I ad ed with P? Solutio : Give word is INDEPENDENCE total letters = Letter Appearace (times) N E 4 D I P C the required umber of arragemets!!4!!!!! = 66 Permutatios ad Combiatios / 67

70 (i) P Box P i Box places remai The required of words startig with P!!!4! = 86 ii) I Eglish laguage, total letters = 6 total umber of vowels = 5 (a, e, i, o, u) Now i give word, Number of vowels are, E 4 times ad I times!! We wat, all vowels always occur together. EEEEI ) we treats them as sigle object; for this situatio; but arrage 5! 4!! ways. 5 = 7 remaiig objects (letters) 7 + = 8 Now but i 8 object N limits & D times The required arragemet iv) 8!!!! x 68 5! 4! iii) The required umber of arragemets = total umber of arragemets (without restrictio) - the umber of arragemets where all the vowels occur together = = I P Box Remaiig places = - = Remaiig letters times N E 4 D C Mathematics & Statistics / 68

71 Hece, the required umber of arragemets!!4!! = 6.!! 4.5 Combiatio Cosider a problem of coutig the umber of subsets of A, such that each subset cotais r elemets ad the set A cotais elemets. We observe that each permutatio of elemets of A, take r at a time ca be produced by performig the followig two tasks i a sequece. Task : Choose a subset B of A cotaiig r elemets of A. Task : Choose a particular permutatio of B. Let x be the umber of ways of choosig B. Task ca be performed i x ways; ad task ca be performed i r! ways. The the total umber of ways of performig both the tasks which is P r, ad by the multiplicatio priciple it is x r! Hece x r!= P r =!r! x =! r! r! Whe r elemets are chose without referece to the order i which they are selected we get a combiatio. Total umber of r combiatios out of thigs i deoted by C r ad has formula : C r = P r r!! r! r! We have the followig results regardig combiatios. (i) C r = C r (usually called as complemetary combiatios) verbally selectig r thigs out of give thigs is equivalet to rejectig ( r) thigs.! Proof : R.H.S. = C r = [by formula] ( r)!( r)! (ii) C = =! r!r! = C r Selectig thigs out of give thigs ca be doe oly i way. (iii) Co = Selectig othig out of give thigs ca also be doe i oly oe way. (iv) C r + C r =(+l)c r Proof : L.H.S. = = = r! r!!! r! r! r!! r r r!!( ) ( )! r!( r)! r!( r)! = (+l)c r Permutatios ad Combiatios / 69

72 We shall illustrate the use of these results i the followig examples : Example () : Fid r, if C r = C r Solutio : By complemetary combiatios, C r = C _ r By data C _ r = C r r = r 4 = 4r r = 6 Note : C r = IC r may give r = r ad r = which is absurd. Example () : Fid, if C 4, C 5 ad C 6 are i A.P. Solutio : By data C 4 + C 6 = ( C 5.)!!! i.e. 4! 4! 6! 6! 5! 5!!! (!) i.e. 4!( 4) ( 5) ( 6)! 6 5 4!( 6)! 5 4!( 5) ( 6)!! Cacelig 6!4! through out, we get ( 4) ( 5) L.C.M. is ( 4 ) ( 5 ) ( 4)( 5) = ( 4) i.e = 48 i.e = i.e.( 4 ) ( 7 ) = = 7, 4. Example () : Fid r, if 4 C s +4 C 6 +5 C 7 _ +6 C 8 = 7 C r. Solutio : We use the idetity C r + C r = ( + l)c r Cosider, 4 C C 6 = ( 4 + ) C 6 = 4, r = 6 = 5C 6 Next, 5 C C 7 = ( 5 + ) C 7 = 5, r = 7 = 6C 7 Fially L.H.S. = 6 C, + 6 C 8 = (6 + ) C 8 = 6, r = 8 = 7C 8 By data 7 C 8 = 7 C r r = 8 or, by usig complemetary combiatio 7 C 7 8 = 7C r i.e. 7C 9 = 7C r r = 9...(A)...(B) Mathematics & Statistics / 7

73 Thus, from (A) ad (B) we get r = 8 or 9 Example (4) : From a pack of 5 playig cards, a had of 5 cards is draw. Fid i how may ways such a had ca be draw. Solutio : Out of 5 playig cards a had of 5 cards is draw at radom. This ca be doe i 5 C 5. ways. = = 5P5 5! = = ways. Example (5) : A valid computer password cosists of four character, the first of which is chose from the set A = {p, q, r, s, t} ad the remaiig three characters are chose from the Eglish alphabet or digits chose from the set T = {,,,, 4, 5, 6, 7, 8, 9}. How may differet passwords are there? Solutio : A password ca be costructed by performig the followig two tasks T ad T i successio. Task T : Choose a startig letter from the set A. Task T : is allowed) Choose a sequece, three characters of letters or digits (repetitio Task T ca be performed i 5 C =5 ways. Sice there 6 alphabets ad digits that ca be chose for each of the remaiig characters, ad sice repetitio is allowed T, ca be performed i (6) = ways. By the multiplicatio priciple, umber of differet passwords is 5(45656) = 8. Example (6) : How may seve-perso committees ca be formed each cotaiig three female members from a available set of female ad four male members from a available set of males? Solutio : A committee ca be formed by performig the followig two tasks i successio : Task T : Task T : Choose female member from a set of females. Choose 4 males from a set of males. Here order does ot matter i idividual choices, so we are merely coutig the umber of possible subsets. Thus task T ca be performed i C 9 8 = = 4 ways. ad task T ca be performed i C 4 = =745 ways. 4 By the multiplicatio priciple, there are (4) (745)=, 4, 7 differet committees. Permutatios ad Combiatios / 7

74 Check your progress : 4.5 () Give that 5 Cr = 5 C r+ fid r C 4. () Prove that 5 C + 5 C + 5 C + 5 C 4 =. () Fid, if C 6 : ( ) C = : 4. (4) A perso has frieds of whom 8 are relatives. I how may ways ca he ivite 7 guests so that 5 of them are relatives? (5) A ur cotais 5 balls, eight of which are red ad 7 are black. I how may ways ca five balls be chose so that (a) all five are red, (b) all five are black, (c) two are red ad three are black, (d) three are red ad two are black. (6) Fid the total umber of (i) rectagles, (ii) squares o a chessboard. (7) A perso has 4 Eglish ad 6 Marathi books. He wats 5 of these to be boud. I how may ways ca he select the 5 books so as to iclude (i) exactly Eglish books (ii) at least Eglish books (iii) at most Eglish books? (8) Prove that C r = r r (C r ) 4.6 SUMMARY Permutatios ad Combiatios meas the proper arragemet of give thig. If there is some coditios the it is permutatio otherwise combiatio. Formula for Permutatio is i Factorial Notatio. We ca use these formula for solvig examples. 4.7 CHECK YOUR PROGRESS - ANSWERS 4. () {H, T} =, 4 sequeces. () 4 6 = 496 sequeces. () = 54 ways. (4) = 6 ways. 4. ad 4.4 ()! () (i)! = 479,, 6 (ii) (6!) =, 6, 8 () 5! = (4) 5! = (5) = 8 (6) r = 4 (7) (i) 7! = 8 (ii) 8! 7! = 4 (8) 8 Mathematics & Statistics / 7

75 4.5 () 5 () (4) 8 C 5 4 C = 6 (5) (a) 8 C 5 = 56 (b) 7 C 5 = (c) 8 C 7 C = 98 (d) 8C 7C = 76 (6) (i) (6 ) = (6) = 96 (ii) 4 (7) (i) 4 C 6 C = 48 (ii) 4 C 6 C + 4 C 4 6 C = 496 (ii) 6C 5 + 6C 4 4C + 6C 4C + 6C 4C = 584 (8)! Cr ( - v!r!)! Cr ( r )!( r )! x (r) ( - r)!( - r (r -)! ( r) x (! r!) ( r ) r Crx ( h r ) Cr r r Hece the proof. Cr! ) Evaluate ) 7! 4.8 QUESTIONS FOR SELF STUDY! ) 6!6! ) Fid the umber of Permutatios obtaied by arragig all letters of the word COMBINATION ) From a group of studets, how may committees ca be formed cosistig of either, or 4 studets? 4) I how may ways, cards of same suit ca be take from a pack of playig cards? 5) Fid the value of if C 4 = 5.p 4.9 SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby Permutatios ad Combiatios / 7

76 NOTES Mathematics & Statistics / 74

77 CHAPTER 5 Liear Equatios 5. Objectives 5. Itroductio 5. Determiat 5.. Determiat of rd order 5.. Cramer's rule 5.. Cosistecy of equatios 5. Matrices 5.. Types of matrices 5.. Algebra of matrices 5.4 Liear homogeeous equatios 5.5 Liear o-homogeeous equatios 5.6 Summary 5.7 Check your Progress Aswers 5.8 Questios for Self Study 5.9 Suggested Readigs 5. OBJECTIVES After studyig the basic cocepts such as determiats, matrices, Cramer s rule you ca very well use the followig solvig homogeeous equatios usig Cramer's rule solvig o-homogeeous equatios testig cosistecy of equatios perform additio, subtractio ad multiplicatio operatios with matrices. 5. INTRODUCTION Before we begi our discussio with liear equatios, we shall cosider certai mathematical tools like determiats ad matrices i brief. These cocepts are very useful i solvig the geeral liear equatios. At the outset we shall restrict ourselves to the case whe the geeral liear equatios have a uique solutio. 5. DETERMINANT The four elemets a, b, c ad d arraged i two arrays cotaiig two elemets ad havig a defiite value is called a determiat of secod order. e.g. a c b d is a determiat of d order, It has two rows a b ad c d. Also it has two colums c a ad d b. We measure rows from top to bottom ad measure colums from left to right. (ad bc) is called the value or expasio of a determiat Liear Equatios / 75

78 Mathematics & Statistics / 76 Thus bc) (ad row row d b c a d st st colum d colum We cosider some simple examples. Examples : Evaluate the followig : () 4 4 = (4) ( ) (4) = 8 + =. () x a = a ( ) () ( x) = x a.. () 4 = () (4) = Determiat of rd order A determiat of rd order has arrays each cotaiig elemets. Thus it has rows ad colums. The expasio of a determiat of rd order is give by b b a a c c c a a b c c b b a c c c b b b a a a =a (b c b c ) b (a c a c ) + c (a b a b ) Examples : Evaluate the followig determiats (4) = () ( ) = (5) x x x x x x x = x (x ) + ( x) = x x + (6) = ( ) (4 4) + (6 6) = 5.. Cosistecy of Equatios : So far we have see the method to solve two simultaeous' equatios i two ukows ad three simultaeous equatios i three ukows by usig Cramer's rule. Now cosider a system of three equatios i two ukows. a x + b y + c = a x + b y + c = a x + b y + c = It is ot always possible to solve these equatios ad get the values of x ad y.

79 If these equatios have a commo solutio we say that the equatios are cosistet. The coditio that these equatios are cosistet is a a a b b b c c c = Whe the coditio is fulfilled, ay two of the above equatios ca be solved to get values of x ad y. i.e. solutio of the system. Cosider a example, which will give us the method. Example (7) : Test whether the followig equatios are cosistet. x + y + 4 =, x + y + =, x + 4y + 5 = If they are cosistet solve them. Solutio : Cosider, the determiat for the cosistecy coditio. 4 D = 4 5 Perfor mi g R R R D = R = = = i.e. D = 4 The give equatios are cosistet. To solve them, we rewrite the first two equatios as x + y = 4 x + y = By Cramer's rule, we have x= y= 4 Hece the equatios are cosistet ad has the solutio x =, y =. Example (8) : Solve the followig equatios by Cramer's rule x + 4y 7 =, 7x y 6 =. Solutio : We shall rewrite the equatios i the form By Cramer's rule, we get 7 x + 4y = 7 7 x y = x = Liear Equatios / 77

80 y = Hece x =, y = is the solutio. The same Cramer's rule ca be exteded to three liear equatios i three ukows. Cosider, a system of liear equatios a x + b y+ C z = d a x + b y + c z = d a x + b y + c z = d The solutio of the system is give by D D x y D z x, y, z if D. D D D where D = a a a b b b c c c D x = d d d b b b c c c D y = a a a d d d c c c D z = a a a b b b d d d The example solved below will illustrate the techique ivolved. Example (9) : Solve the followig system of liear equatios x + y + 4z 4 =, x + y + 6z 5 =, y + z x + 4 =. Solutio : The give system of equatio ca be rewritte as x + y + 4z = 4 x + y + 6z = 5 x + y + z = 4 Usig Cramer s rule we solve the give system of equatio Dx = D = - +4 = (x x6) - (x-(-)x6) +4 (x-(-)8) = (-) - (+8) + 4 (4+9) =x-9 x +4 x = = = Mathematics & Statistics / 78

81 = = 4 (x x6) - (5x (-4) x 6) +4 (5x (-4)x) = 4 (-) - (5+4) +4 (-) = 4x 9 - x 9 + 4x = = 4 4 Weget, Dy = Lastly Dz = = = (5 x (-4) x 6) -4 ( x (-) x 6) +4 ( x (-4) (-) x5) = (5 + 4) -4 ( + 8) +4 (-8 + 5) = x 9 4 x + 4 x 7 = = = = ( x -4 5 x ) - (x 4 (-) x 5) +4 (x- (-) x ) = (--) -(-8+5) + 4(4 + 9) = x - x x = = Now by Cramer s Rule, we have, Dx x D Dy y D Dz z D Hece solutio i (x, y, z) = (, -, ) 5.. Cramer's rule Cosider the system of two simultaeous liear equatios i two ukows. a x + b y = c a x + b y = c The solutio of these equatios is give by Cramer's rule as x = x = c c a a c b a b b b b b c a b b y = y = Liear Equatios / 79 a a a a a c a b c c b b a a We shall illustrate this by meas of a simple example. c b

82 Check your progress 5. () Fill i the blak by choosig correct value from the bracket (i) = (,,4, 4) 4 (ii) = (,,, 7) (iii) 5 7 =... (,5,7,) (iv) If x + 5y = 5 ( x =, y = ; x = y = ; x =, ad x + 7y = 6 (v) If y = ; x =, y=l) the x =..., y =... x y ad x =..., y =... x y 4 (x = 4, y = 4; x = 4, y = 4 ; x =, y = ; x =, y = ) () Test for cosistecy the followig equatios. If foud cosistet, obtai the values of x, y. (i) x + y = 4 (ii) 4x y + = () Show that x + 4y = 7 7x 8y + = llx+y = 7 x + y 5 = a b c (i) b c a c a b (ii) (b + c) x + ay + = (c + a) x + by + = (a + b) x + cy + = are cosistet. 5. MATRIX We shall ow cosider aother etity called a matrix, which has umerous applicatios i almost every field. Defiitio : A matrix is a rectagular array of umbers arraged i m rows ad colums as Mathematics & Statistics / 8

83 a A= a... a ml a a a... m a I a... a m The i th row of A is [ a i., a i..... a i ] ( i m) ad j th colum of A is a a : a j j mj ( j ) The order of matrix A is said to m, as it has m rows ad colums. The elemet commo to i th row ad j th colum is a.ij.. It is (i, j ) th etry of A. We briefly write A = ( a ij ) m 4 e.g. (i) A = is a matrix of order. (ii) B = is a matrix of order. 4 (iii) C = 4 is a matrix of order. 7 The elemets a, a, a...a form the pricipal diagoal of a matrix (a. ij.) a + a, + a a is called the trace of the matrix A = (a.ij.) 5.. Types of Matrices : We shall ow cosider certai types of matrices. () Row matrix : It is a matrix havig a sigle row. e.g. [ 4 5] l 4 () Colum Matrix : It has a sigle colum. e.g. is a colum matrix of order 4 4 () Null Matrix : It is a matrix of ay order, whose all the elemets are zero. (4) Square Matrix : It is matrix havig umber of rows ad umber of colums equal. 4 e.g (i) is a square matrix of order or rowed square matrix. a h g (ii) h b f, g f c are square matrices of order. Liear Equatios / 8

84 Mathematics & Statistics / 8 (5) Diagoal Matrix : It is a square matrix whose o-zero elemets are alog the diagoal a ii e.g. 4 = diag(4,), = diag (,, ) are diagoal matrices of orders, respectively. (6) Scalar Matrix : It is a diagoal matrix havig all elemets equal. a ii = k for i = j e.g. diag (k k k) = k k k (7) Uit Matrix : It is a scalar matrix, havig all diagoal elemets equal to. eg. I =, I = (8) Triagular Matrix : It is a square matrix i which all the elemets above or below the pricipal diagoal are zero. e.g. A= B= 8 are both triagular matrices. (9) Sigular Matrix : It is a square matrix whose determiat is equal to zero. e.g. (i) A = 6 4 Cosider, A = 6 4 =4 4 =. (ii) B= Cosider, B = Performig R R, we get B = 5 4 = because R. = R, B = Thus both matrices A ad B are sigular. () No-sigular matrix : It is a square matrix, whose determiat is o-zero. e.g. (i) C =, Cosider, C = =+6 = 7 C. (ii) D = 5 5 4, Cosider, D = Performig R 4 R, R + R we get

85 Liear Equatios / 8 D = 8 = 8 = 4 D =. Thus both C ad D are o-sigular matrices. 5.. Algebra of Matrices : We shall briefly cosider the followig operatios with matrices : (I) Traspose of a Matrix : It is a matrix obtaied by iterchagig rows ad colums. The traspose of A is deoted by A or A T. Let A = ( a ij ) m the A' = A T = ( a ji ) m e.g. A = 5 4 4, A = A T = are trasposes of each other. (II) Multiplicatio by a Scalar : Let k R ad A = (a. ij ) m - The ka = (ka ij ) m i.e. every elemet of the matrix A is multiplied by the scalar k. e.g. A = 5 the A = ) 5 x x x x x x (III) Equality of Matrices : Two matrices A ad B of the same order are said to be equal, if the correspodig elemets are equal. i.e. A = (a. ij.) m B = (b ij.) m A = B, if ad oly if a ij. = b ij. for all i, j. (IV) Additio of two matrices : Two matrices of the same order ca be added. Let A : = (a ij ) m ad B = (b ij ) m A + B = (a ij + b ij ) m x i.e. correspodig elemets of the two matrices are added. e.g. A= 7 4 B = A + B= ) 4 4 (V) Subtractio of Matrices : Two matrices of the same order ca be subtracted. A B = (a ij.. b ij ) m Refer to matrices A,B i.(iv).above. A B = (VI) Multiplicatio of Matrices : Let A = (a ij ) m ad B = (b jk ) p be two matrices. The product AB will be defied if the umber of colums of A = umber of rows of B. I the above case AB is defied ad is of order m p. Let C = (c ik ) m p = AB

86 The, c ik = a ij b jk = a ib k + a i b +. k... + a i b k j e.g. A = 4, B = c The product AB is defied ad is of order. Let AB = C = c c = ()( )+( 4)+4( )= c = ()+(5)+4(6)=45 c = ( )( )+( )( 4)+()( )= c = ()+( )5+(6)= AB = 45 Here the product BA is also defied ad is of order. Let BA = d = ()+( l)= 7 d = () + ( )= d = (4)+()= 8 d = 4() + 5( l)= d = 4() + 5( )= d = 4(4) + 5()= 6 d = l() + 6( l)= 8 d = l() + 6( )= 5 d = l(4) + 6()= 4 BA = = = D= (d ij ) Note : Eve though AB ad BA are both defied, they are of differet order. Hece AB BA. c c Mathematics & Statistics / 84

87 Liear Equatios / 85 Check your progress : 5. () Fill i the blaks by choosig appropriate type of a matrix (i) ( 4) is a matrix. (ii) 4 7 is a matrix. (iii) is a matrix. (iv) is a matrix. (v) is a matrix. (vi) a a a is a matrix. () State whether true or false. (i) (ii) (iii) 7 4 ad ca be added / subtracted. (iv) b 4 a 4 the a =, b = (v) A= 4, B = 4 5 the AB = 4 4 (vi) ad are compatible for multiplicatio. 5.4 LINEAR HOMOGENEOUS EQUATIONS Cosider a system of m equatios i ukows viz. x r x,. x a x + a x + a x +..+ a l x = a x + a x + a x + + a x = : :

88 a m x +a m x +a m x +.+a m x = The system ca be writte i a matrix form as AX = where A= a a... a ml x x ad X = : x a a a... m a a a... m m ad O= : A is of order m, X ad O are colum matrices of order each. I geeral such a system has a trivial solutio. i.e. x = = x =..= x is a solutio. Uder certai coditios a system may have a o-trivial solutios. Examples : () Solve the followig system of equatios ad obtai a set of ozero solutios; if possible x + y + z = x + y z = 5x + 4y + z = Solutio : The matrix equivalet of the system is AX = B x y 5 4 z [Note : We may use elemetary trasformatios to reduce the above matrix equatio. Oly row trasformatios are permissible.] (i) Iterchage of i th row / colum with i th row / colum ad is deoted by, R. ij / C ij.. (ii) (iii) Multiply elemets of (i th ) row / colum by a scalar k ad is deoted by, k(r i )/k(c i.) Multiply elemets of i th row / colum by k ad addig it to the correspodig elemets of j th row / colum ad is deoted by, R j. + k(r i.)/c j. + k(c i ) ] Performig R + R ad R R we get 4 4 Here o-zero solutio does ot exist. The determiat of the coefficiet matrix is x y z 4 4 = = 9 6= i.e. The coefficiet matrix is o-sigular. Hece oly solutio is give by x+y+z =, x+4y =, 4x+y = Thus system has oly a trivial solutio. x = = y = z Example () : Obtai a set of o-zero solutios. x + y + z = Mathematics & Statistics / 86

89 Liear Equatios / 87 x + y z = x + y + z = Solutio : The matrix equivalet of the system is z y x Performig R R R ad R R, we get 7 z y x The system reduces to equatios x + y + z = ad y + 7z = y = 7 z ad x = y z = 4 z z = z 5. Let z = k. The x = 5k ad y = 7k. Thus for ay o-zero value of k, we get ifiitely may solutios of the form x = 5k, y = 7k ad z = k Example () suggests aother method to get ifiitely may solutios i case of two equatios i three ukows. Let a x + b y + c z = ad a x + b y + c z = Solvig these for proportioal values of x, y, z we get b a b a z a c a c y c b c b x i.e. b a b a z a c a c y c b c b x Example () : Solve for proportioal values (or obtai ifiitely may solutios of) x + y + 4z = x + y + 6z = Solutio : Writig the solutios i the determiat form, we get z 6 4 y 6 4 x i.e. z 4 y 4 8 x i.e. 5 z 6 y 4 x = k say Solutio is x = 4 k, y = 6 k, z = 5k, k R.

90 Mathematics & Statistics / 88 Check your progress : 5.4 Solve the followig system of equatios by matrix method: () x + y + 4z =, x + y + z=, x + y + z = () x+y z=, x+4y+z=, 4x+6y= () x+y+z=, x+y+z= (proportioal values oly) 5.5 LINEAR NON-HOMOGENEOUS EQUATIONS Cosider a system of m equatios i ukows viz. x, x,.., x a ll x l + a x + a x +..+ a x = b l a x l + a x + a x a x = b a ml x l + a m x + a m x + + a m x = b m The system ca be reduced to a matrix form AX = B where A = ( a ij ) m x, X = x : x x, B= m m b : b b If the coefficiet matrix is o-sigular the the solutio exists ad is uique. We shall illustrate this cocept by the followig examples. Example () : Solve the followig system of equatios by matrix method, x + y+ z = 6 x y + z = x y +z = Solutio : The matrix equivalet of the system is 6 z y x i.e. AX = B cosider, A = Performig R R, R + R, we get A = 6 Here A A is o-sigular. System has a uique solutio. By usig elemetary row trasformatios, we ca reduce A to a triagular matrix. We have 6 z y x

91 Liear Equatios / 89 Performig R R, R + R z y x i.e. x + y + z = 6, y z = 9, z = 6 z =, y = 9+z = 6 y = ad x = 6 y z = 6 = l Hece the system has a uique solutio give by x =, y =, z =. Example () : Solve the followig system by matrix method, x + y z = x y + z = 5 5x y + z = Solutio : The matrix equivalet of the system is 5 z y x 5 i.e. AX=B The coefficiet matrix A = 5 Cosider, A = 5 Performig R R, R 5 R, we get A = = 4+ = Hece coefficiet matrix is o-sigular. The system has a uique solutio. We have 5 z y x 5 Performig R R, R 5 R, we get 9 z y x 8 7 Performig, R we get z y x 8 7 Now R 7 R gives

92 Mathematics & Statistics / 9 z y x i.e. x + y z=, y + z =, z = z =, y =, x = The solutio is x =., y =, z =. Example () : Solve the followig system of equatios x y + z= x + y + z = x + z = Solutio : The matrix equivalet of the system is z y x i.e. AX = B; where A = Cosider, A = Performig R R R R, we get A = Performig R R we get = = The matrix A is sigular. Hece the solutio is ot uique. Cosider, the system z y x Performig R R R R, we get z y x Performig R R ad R, we get z y x

93 i.e. x y + z= ad y z = The system has ifiitely may solutios. Let y = k z = k l ad x = +y z= l+k k + = k The solutio is x = k, y = k, z = k, k R. By givig differet values to k, we get ifiitely may solutios. Check your progress 5.5 Solve the followig system of equatios, () 4x + y 4z = x y + z = x y = 6 () x + y + z = 5 y z = 5 6x + y z = () x + y + z = x y + z = 6 x y + z = SUMMARY Liear Equatios are the mathematical relatioship betwee variables. Degree of this equatio is oe so it is called as liear equatio. There are oe, two, three, ay umber of variables you ca fid values for these variables. To solve system of equatio we have Cramers Rule, Matrix Method CHECK YOUR PROGRESS - ANSWERS () (i) 4 (ii), (iii) (iv) x =, y = (v) x = 4, y = 4 () (i) icosistet (ii) cosistet, x =, y = () (i) row (ii) colum (iii) diagoal (iv) uit (v) sigular (vi) scalar () (i) true (ii) false (iii) false (iv) true (v) true (vi) false () x =, y =, z = () x = k, y = k, z = k, k R { } () x = k, y = k, z = k, k R { } () x=, y =, z = () x =, y =, z = () x= k, y =, z= k k R Liear Equatios / 9

94 5.8 QUESTIONS FOR SELF - STUDY ) Fid the value of a, b, c, d from followig equatios a b 5c d a b 4 = 4c d 4 ) If A = B = Calculate : AB, BA ad check AB = BA. 5 the 4 ) Let A = B = C = 5 Fid a) A+B b) A B c) A C 4) Fid Value of x. = x SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby. Statistical Aalysis: A Computer - Orieted Approach Itroductio to Mathematical Statistics by S. P. Aze & A. A.Afifi Mathematics & Statistics / 9

95 NOTES Liear Equatios / 9

96 NOTES Mathematics & Statistics / 94

97 CHAPTER 6 Quadratic Equatios 6. Objectives 6. Itroductio 6. Complex Numbers 6. Solutio of a quadratic equatio 6.4 Nature of roots 6.5 Quadratic equatio with give roots 6.6 Summary 6.7 Check your Progress - Aswers 6.8 Questios for Self - Study 6.9 Suggested Readigs 6. OBJECTIVES After studyig the cocept ad operatios with complex umbers you ca very well make use of the followig : solvig the quadratic equatio discussig the ature of roots formatio of a quadratic equatio relatio betwee roots ad coefficiets of the quadratic equatio. 6. INTRODUCTION We have see a liear expressio i oe variable viz ax+b. Similarly a quadratic expressio i oe variable is ax + bx + c (a ). By equatig to zero, a quadratic expressio, we get a quadratic equatio. viz ax +bx+c =. To solve such a equatio, we shall express the quadratic expressio as a differece of two squares ad the factories it. Before we come to the solutio of such a equatio we shall defie complex umbers ad study its elemetary operatios. 6. COMPLEX NUMBERS We defie positive square root of by i i.e. i = + so that i =. i is called imagiary uit. If a, b are real umbers that a + ib is called a complex umber. a is called real part ad b is called imagiary part of the complex umber (a + ib). The umbers (a + ib) ad (a ib) are called complex cojugates of each other. The real umber a ib. a b is called modulus of (a+ib) ad is deoted by Also a ib = a. a b = ib For example, i,( ) i,4 i Are complex umbers. For the complex umber Z = a+ ib, a is called the real part, deoted by Rez ad b is called imagiary part deoted by Imz of the complex umber z. Quadratic Equatios / 95

98 For example, If Z = +5i The Rez + ad Imz = 5. The two complex umbers Z = a + ib ad Z = a ib Are called as complex cojugate of each other. Where Z = a + ib Ad Z = a ib = Z Let Z = a + ib ad Z = C + id the these two complex umbers are said to be equal if a = c ad b = d 6.. Algebra of Complex Numbers (I) Additio Let Z = a + ib, Z = C + id be ay two complex umbers. The sum (additio) is defied as follows : Z + Z = (a + ib) + (C+ id) For example Z = + i ad Z = 5 i The = (a + c) + i(b + d) Z + Z = ( + i) + (5 i) = ( + 5) + ( ) i = 7 + i = (7 + i) (II) Subtractio Z = a + ib Z = C +ib the Z Z = (a + ib) (C + ib) e.g. Z = + i, Z = 5 + i = (a c) + (b d)i the Z Z = ( + i) (5 + i) = ( 5) + ( )i = (-) + i = (-+i) (III) Multiplicatio Z = (a + ib), Z = (C + id( The Z x Z = Z Z For example, = (a + ib) (c + ib) Z = + i5, Z = + 6i = (ac bd) + (ad + bc)i The Z Z = ( + 5i) ( + 6i) = (+6i) + 5i(+6i) = x + x 6i + 5i x + 5i x 6i Mathematics & Statistics / 96

99 = 6 + 8i + i + i C = 6 + 8i + x i = 6 + 8i = i (IV) Divisio Z = a + ib, Z = C +id the but Z Z a ib c id We have multiply umerator ad deomiator by complex cojugate of Z as Z = C id Z Z a ib c id x c id c id ( a ib)( c id) ( c id)( c id) a( c id) ib( c id c( c id) id( c id) ac iad ibc i bd c cid cid i d ac i( bc ad bdx c i d ac bd) i( bc ad) c d ac bd bc ad i c d c d (V) Modulus Value of Complex Number Let Z = a +ib the Z = a + ib = ad a b Complex Cojugate of Z is Z Ad Z = a ib The z = a+ib = a ( b) a z z a b b For example Z = a+i, Z = +5i the fid the modules values of give complex umbers Solutio Give Z = 9 + i z = ( 9) () = 8 4 = 85 Quadratic Equatios / 97

100 Ad Z = -5i the Z = ( 5) = 9 5 = 4 Example () : Express the followig i the form a + ib. (i) ( + i) + (4 7i) = ( + 4) + i( 7) = 6 i (ii) ( + i) + ( i) = ( + ) + i( ) = 4 + i() (iii) (4 + 5i) (7 i) = (4 7) + i(5 + ) = +7i (iv) (4 + 5i) (4 5i) = (4 4) + i(5 + 5) = + i (v) (l + i)( i) = (l)() + i (l)( l) + i i = 4 + i (vi) (l+i) ( i) = l i = (vii) i i ( i) ( i) 6 i i = ( i)( i) 4 i i = 5 5i = +i 4 5 Example () : Fid the modulus of the followig umbers. Solutio : (i) ( + i)( 5i) (i) ( + i)( 5i) = 6 5i + i i ( i)( - 5i) = 7 i = i = 7 i (ii) 7 7 i i = = (ii) 7 i (7 i) (l i) 7 i 7 i i = i ( i) ( - i) i 7 i i = 6 8i = +4i = 4i = 9 6 = 5 = 5. Here is real part ad 4 is imagiary part of ( + 4 i). Example () : Factorise : x + x + 5. Solutio : We have x + x + 5 = x + x + l+4 = (x+l) ( 4) = (x+l) ( l)() i Mathematics & Statistics / 98

101 = (x+l) (i) a b ( a b)( a b) = (x + l+i)(x + l i). Check your progress 6. () Fill i the blaks. (i) the real part of ( 7 i) is (ii) the imagiary part of ( i) is (iii) The cojugate of (7 i) is (iv) the sum of the complex umbers ( 5 i) ad ( + 5 i) is (v) the differece ( + i) ( i)is (vi) the product of ( + i) ad ( 7 i) is i (vii) the ratio of is i (viii) the modulus of ( i) is () Express as a complex umber with real deomiator ad state its real ad i imagiary part of i () Factorise : x + x SOLUTION OF A QUADRATIC EQUATION We kow that a quadratic equatio is of the form ax +bx+c= (a ) Divide by a. x b c = a a b b c b i.e. x x = a a a 4a b b 4ac i.e. x = a 4a b b 4ac i e x = a a i e x b a b 4ac a x b a b 4ac a = b roots are x = Let us deote these roots by,. The the sum of the roots is b 4ac, a b b a 4ac + = = b ad the product of the roots is b 4ac b a b b = a a b a 4ac coefficiet of x coefficie t of x Quadratic Equatios / 99

102 = b b 4ac a b b a 4ac = = b b b 4ac 4a b 4ac 4a c a Cos tatterm coefficiet of x = Thus + = b a ad c a We shall illustrate the use of formulae derived, by the followig examples. Example () Solve the followig equatios (i) 6x x 6 = (ii) 6x 5x + = (iii) 9x + 6x + = (iv) 56x 9x + 85 = Solutio : (i) O comparig 6x x 6 = with ax + bx + c =, we get a = 6, b =, c = 6. b The solutios are x = b a 4ac Here x = x = ad x = are the roots. (ii) we have a = 6, b = 5, c =. Give 6x 5 Comparig coefficiets with ax bx c We have x = x = ad x = (iii) we have a = 9, b = 6, c =. Give ax 6x Comparig with ax bx c are the roots x = Here both the roots are equal to Mathematics & Statistics /

103 (iv) here a = 56, b = 9, c = 85. The solutios are Give 56x 9x 85 Comparig with ax bx c We have The solutio are b b 4ac x a As 9 z x = = = i i 6 7i Here the roots are, 6 6 i.e. a pair of complex cojugates. Example () : If ad are the roots of the equatio x 5x +7 =, evaluate the followig : (i) +β (ii) α β (iii) α β α α Solutio : We have, β are roots of x 5x + 7 =. Comparig with ax bx c, b 5 Here a =, b = 5, c = 7 + β =, a β = a c = 7. (i) (ii) (iii) + β = ( + β) β 5 x Quadratic Equatios /

104 Check your progress 6. () Solve the followig equatios (i) x + 4x + 4 = (ii) x 4x + = (iii) x + 5x + 8 = (iv) x + 5x 444 = () Give that β are the roots of x + 5x =. Fill i the blaks : (i) + β =... (ii) β =... (iii)... (iv)... (v) ( + l) (β + l) = NATURE OF ROOTS I example () of the previous article, we have see that the roots of the quadratic equatios are of four types viz (i) real ad uequal (ii) ratioal ad uequal (iii) equal (iv) pair of complex cojugates. All these roots deped o the value of b 4ac. We call b 4ac as a discrimiat of ax + bx + c = ad deote it by. We will classify the roots as above ad write dow the ecessary coditio i terms of. (i) (ii) real ad uequal roots : ratioal ad uequal roots : a, b, c, are all ratioal ad is a perfect square. (iii) equal roots (each equal to b ) : = a (iv) a pair of complex cojugate roots : < Example () : Determie the ature of the roots of the followig equatios : Solutio : (i) x + x+ 4 9 = (ii) 8x 9x+8 = (iii) 9x + 9x 4 = (iv).x.4x + = a Give x x 4 (i) O comparig x + x = with ax + bx + c =, we have a =, b =, c = 9, 4 = b 9 4ac = 9 4 ( ) = 4 Hece the roots are equal ad each root is = (ii) Give 8x 9x 9, Here a = 8, b = 9, c = 8 b a. Mathematics & Statistics /

105 = b 4ac = ( 9 ) 4 ( 8 ) ( 8 ) = ( 9 ) ( 6 ) i.e. = ( 5 ) > The roots are real ad uequal. Give (iii) Here a = 9, b = 9, c = 4 = b 4ac = 8 4 ( 4 ) ( 9 ) = = 5 = ( 5 ) Thus is a perfect square, ad a, b, c are ratioal. Hece roots are ratioal ad uequal. Give (iv) Here a =., b =.4 ad c = = b 4ac = (.4 ) 4 (. ) ( ) =.96.4 =.44< Hece roots are complex cojugates of each other. Example () : Fid the value of k i each of the followig, give x + 6kx + k + = is a quadratic equatio. (i) The sum of roots is zero. (ii) The product of roots is zero. (iii) Oe root is reciprocal of other. (iv) Both roots are equal. (v) The roots are complex cojugates of each other. Solutio : Let, β be the roots. Here a =, b = 6k, c = k + The + β = b a ad β = 6k = c k a (i) + β= gives k = k =. (ii) β = gives k = k = = k (iii) If oe root is reciprocal of the other root, product of the roots =. k β = k =. k + = (iv) If both roots are equal, = b 4ac ad = b 4ac = (v) the 6k 4()(k + ) = i.e. k (k + ) = i.e. k k + k = i.e. (k + ) (k ) = k = or k = The roots are complex cojugates of each other. = b 4ac < i.e. (k + )(k l)< Case (i) k+ >, k < Quadratic Equatios /

106 i.e. k > i.e. ad k < < k <. Hece roots will be complex cojugates, if <k. Case (ii) k + <, k > i.e. k <, k > There is o commo value satisfyig both iequalities. Hece it is ot feasible. Check your progress 6.4 () Fill i the blaks by choosig the appropriate word from (real ad distict, ratioal, irratioal, complex cojugate, real ad equal) (i) The roots of the equatio x + 5x + = are... (ii) The roots of the equatio 9x + x + 4 = are... (iii) The equatio x + x + = has... roots. (iv) The equatio x 8 = has... roots. (v) The roots of the equatio x + x = are... () State whether true or false, if false correct it. (i) Oe root of the equatio x 9x + k + 4 = is the reciprocal of the other, the k = 6. (ii) The sum of the roots of the equatio x 6x + kx k = is zero, the k=. (iii) The equatio x 6x + k = has equal roots, the k = 6. (iv) The equatio x + hx + 8 = has complex cojugate roots, if 4 < h < QUADRATIC EQUATION WITH GIVEN ROOTS If, β are the roots of the equatio ax + bx + c =, we have +β = a b, c a a O Sice divide quadratic equatio by a. b c x x i.e. x b c a a a a i.e. x ( + β) x + β = Hece x (sum of roots) x + product of the roots = This gives us the formula to obtai a quadratic equatio with give roots. Note : (i) if ( + i β) is oe root of the quadratic equatio, the the other root is ( iβ). (ii) If oe root is( + ), the the other root is ( ). We shall illustrate the formulatio of a quadratic equatio by the followig Mathematics & Statistics / 4

107 examples. Example () : Form the quadratic equatio with real coefficiets, oe of whose root is Solutio : (i) (7 + i ) (ii) ( 5 ) (i) By the above ote, the other root is cojugate of (7 + i ). Hece the roots are (7 + i ), (7 i ) Sum of the roots = (7 + i ) +( 7 i ) = 4 ad product of the roots = ( 7 + i ) ( 7 i ) Hece the quadratic equatio is x 4x + 5 =. (ii) The other root is ( + 5 ). = 7 ( i ) = 5. The roots are ( + 5 ), ( 5 ). Sum of the roots = ( + 5 ) + ( 5 )=6 ad product of the roots = ( + 5 )( 5 ) = 9 (5 ) Hece the quadratic equatio is x 6x 66 =. = 9 75 = 66 Example () : Fid the quadratic equatio whose roots are + 5, β + 5, give that β are the roots of the equatio x x + 5 =. Solutio : Sice, β are the roots of x x + 5 =, we have +β = β = x = + = b = a c 5 a Sum of ew roots = ( + 5) + (β + 5) x = + = = (+β) + Product of ew roots = ( + 5)(β + 5) = 4 β+( + β) = 4 5 Quadratic Equatios / 5

108 = 5. Hece the required equatio is x x + 5 =. Example () : If the sum of the roots of a quadratic equatio is 5 ad the sum of their squares is 7, fid the equatio. Solutio : Let, β be the roots of the quadratic equatio. The sum of the roots is 5. + β = 5 The sum of their square is 7 + β = 7. We have β = ( +β) ( +β ) = (5 ) 7 = β = Hece the required quadratic equatio is x ( ) x x 5x =. Check your progress 6.5 () State whether true or false. If false write correct aswer., β are the roots of the equatio x + 5x 4 = (i) The equatio with roots, is x + 5x + 4 = (ii) The equatio with roots, β is x 5x 4 = (iii) The equatio havig roots +, β + is () Fill i the blaks. (i) x x = Oe root of the equatio x + px + q = is β i, the the other root is (ii) Oe root of the equatio x + x + p = is + ad p =, the the other root is 6.6 SUMMARY Quadratic equatio meas a equatio with degree two. There or three variables. We ca calculate value of oe or two ukow variables with give methods of solvig examples of quadratic equatios. There are two roots ad β. There is proper relatio betwee ad β. If value of oe root is give we ca calculate the other root ad also the respective quadratic equatio. 6.7 CHECK YOUR PROGRESS - ANSWERS 6. () (i) (ii) (iii)7 + i (iv) (v) + 4 i (vi) i 5 4i (vii) 5 (viii) i,, () Mathematics & Statistics / 6

109 () x + x + 7 = (x+l+ 6 i ) (x+l 6 i) 6. () (i), (ii) 7 + 4, 7 4 (iii) 6, (iv)47, 5 () 5 (i) (ii) (iii) 5 9 (iv). (v) Hit :( +l) (β+l) = β + ( + β) + l = 6.4 () (i) real ad distict (ii) real ad equal (iii) complex cojugates (iv) irratioal (v) ratioal () (i) True (ii) False, correct : k = (iii) 9 False, correct : k = (iv) True. 6.5 () (i) False, correct : 4x 5x = (ii) True (iii) False, correct : x x = () (i) + βi (ii), p=l. 6.8 QUESTIONS FOR SELF - STUDY ) Fid (a) z z (b) z z (c) z z (d) z z (e) z z Where z = (i+i), z = ( i) ) Express the i form of a + ib ) If i ( i) ( i) i = x + iy fid x, y. 4) Fid Modulas value of z = +i 5) Simplify (+5i) 6.9 SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya ad M. K. Kelkar. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby. Statistical Aalysis: A Computer - Orieted Approach Itroductio to Mathematical Statistics by S. P. Aze & A. A.Afifi Quadratic Equatios / 7

110 NOTES Mathematics & Statistics / 8

111 CHAPTER 7 Probability 7. Objectives 7. Itroductio 7. Defiitios 7. Additio theorem o probability 7.4 Coditioal probability 7.5 Idepedet evets 7.5. Multiplicatio theorem 7.6 The probability model 7.7 Summary 7.8 Check your Progress Aswers 7.9 Questios for Self - Study 7.9 Suggested Readigs 7. OBJECTIVES After studyig the cocept of probability you ca easily solve ad explai problems based o the followig : Additio theorem Coditioal probability Idepedet evets Multiplicatio theorem Probability model 7. INTRODUCTION The theory of probability has its origi i the study of experimets with ucertai outcomes. Oe such popular experimet is tossig a coi or tossig a six faced dice. I case of a coi, we ca almost say that either tail or head will appear. I a six faced dice, we say that ay oe of the six umbers {,,, 4, 5, 6} will appear o top, whe it is rolled. 7. DEFINITIONS () The set of possible outcomes of a experimet is called a sample space, usually deoted by S or X. or Ω. Every outcome is called a sample poit. e.g. (i) whe a coi is tossed, sample poits are H ad T ad sample space is S={H, T} (ii) Whe a six faced dice bearig os. to 6 is rolled, sample space is S={,,,4,5,6}. () Ay subset of a sample space is called a evet. e.g. gettig a odd umber o top of a dice, the evet is = {,, 5} S () Whe the evet is a sigleto set, it is called a simple evet. (4) The etire sample space is a certai evet. (5) Probability fuctio : P : S [, ] If S = {x, x,.. x }the P has to satisfy two coditios. (i) P (x i.) ad (ii) p (x i ) = l. i Thus P(x) is a o-egative fractio lyig betwee ad. Probability / 9

112 e.g. If S is a sample space havig sample poits ad occurrece of all them are equally likely (equi-probable space) the probability of every sample poit is. (6) Probability of a evet is the sum of the probabilities of all sample poits i the evet. e.g. : A pack of 5 playig cards is shuffled ad a card is draw. The 4 probability that it is a kig is. For, let A be the evet that a kig is 5 draw. Sample space cosists of 5 sample poits ad there are 4 kigs. ( S ) = 5, ( A ) = 4. (A) meas umber of elemets i the set A. The probability of every sample poit is. 5 P (A) = Alteratively P(A) = 5 Icidetally this gives us a formula for Mathematics & Statistics / A 4 S 5 Probability of a evet A = P(A) =. A S (7) The complemet of the evet A is deoted by A or ac ad P(A ) = l-p(a). (8) A ad B be two evets. (i) A B meas occurrece of both evets A ad B. (ii) A B meas oe of the evets A ad B may happe or both A ad B happe together or at least oe of the two evets happe. (9) Evets A ad B are mutually exclusive meas if A happes, B does ot happe or if B happes the A does ot happe. e.g. whe a dice is throw, Let A={,, 5} ad B = {, 4, 6} the AB= Thus A ad B are mutually exclusive evets. Also evets A, A are always mutually exclusive. () The empty set is a impossible evet. () Evets A, B, C are exhaustive if, S= AB C Evets A, A are exhaustive evets. () The probability of certai evet S is P(S) = () If A ad B are mutually exclusive evets the probability of evets A or B is P(AB) = P(A)+P(B). We shall illustrate these defiitios by the followig examples : Example () : Three cois are tossed simultaeously. State the sample space ad metio the evets A, B ad A B where (i) A is said to have occurred whe a outcome is heads ad tail. (ii) B is said to have occurred whe a outcome of the experimet is or more heads. 4 5

113 Solutio : The sample space S cotais = 8 sample poits. (S) = 8. S = { HHH, TTT, HTT, THT, HHT, THH, HTH, TTH } (i) A = {HHT, THH, HTH } (A) = (ii) B = {HHT, THH, HTH, HHH } ( B ) = 4 (iii) A B = {HHT, THH, HTH } ( A B) = Example () : Three ubaised cois are tossed. Fid the probability of (i) gettig at least two heads up (ii) gettig at least oe head up. Solutio : The sample space has = 8 sample poits. (i) S = { HHH, HHT, HTH, THH, TTH, TTT, THT, HTT } ( S ) = 8 Let A be the evet of gettig two heads or three heads. A = {HHT, HTH, THH, HHH } (A ) = 4 p(a ) = (iii) A 4 S 8 Let A be the evet of gettig at least oe head up. i.e. A cosists of gettig head, heads or heads. This is complemetary to the evet of gettig o head. Alterative method gettig oe head {HTT, THT, TTH}, Gettig two heads {HHT, HTH, THH}, Gettig three heads {HHH} P( A) 7 8 p A = { TTT } (A ) =. p 8 8 p 8 P(A ) = p(a ) = A S 7 = 8 8 () Two ubaised dice are throw i air. Fid the probability i each of the followig evets. (i) score is a perfect square (ii) score is a multiple of five (iii) score is at most five (iv) score is a prime umber or a perfect square (v) score o each dice is the same (vi) score o the secod dice is greater tha the score o the first dice.. Probability /

114 Solutio : Let x, y be the scores o the st ad d dice respectively. The sample space S ca be show as S = { (x, y) x, y = l,,, 4, 5, 6 } (S) = 6 ad probability of each sample poit is. 6 (i) Let A be the evet that the score is a perfect square. A, = { (x, y) x + y = 4 or 9 }. Score Sample poits Number of sample poits x + y = 4 (, ), (, ), (, ) x + y = 9 (, 6), (6, ), (4, 5), (5, 4) 4 (A ) = 7 ad P(A ) = A 7 S 6 Table 8. (ii) Let A be the evet that the score is a multiple of 5. A = { (x, y) x + y = 5 or } Total 7 Score Sample poits Number of sample poits x + y = 5 (. 4), (4, ), (, ), (, ) 4 x + y = (4. 6). (6. 4). (5. 5} Total 7 (iii) iv) (A ) = 7 ad P(A ) = A 7 S 6 Table No. 8. Let A be the evet that the score is atmost five. A = { (x, y) x + y =,, 4, 5 } Score Sample poits Number of sample poits X +y = (.) x + y = (, ), (, ) x + y = 4 (, ), (, ), (, ) x + y = 5 (. 4), (4, ), (, ), (, ) 4 (A ) = ad P(A ) = Table 7. A 5 S 6 8 Total. Let A 4. be the evet that score is a prime umber or a perfect square Mathematics & Statistics /

115 A 4 = { (x, y) x + y =,,4,5,7, 9, } Score Sample poits Number of sample poits x + y = (,) x + y = (, ), (, ) x + y = 5 (, 4), (, ), (4, ), (, ) 4 x + y = 7 (, 6), (6, ), (, 5), (5, ), (, 4) (4, ) 6 x + y = (5, 6), (6, 5) x + y = 4 (, ), (, ), (, ) x + Y = 9 (, 6), (4, 5), (5, 4), (6, ) 4 Table 7.4 Total P (A 4 ) = A4 S 6 8 (iv) Let A 5 be the evet that the score o each dice is the same A 5 = { (x, y) x = y } A 5, = { (, ), (, ), (, ), (4, 4), (5, 5), (6, 6) } (A 5 ) = 6 A P (A 5 ) = S (vi) Let A 6 be the evet that the score o the secod dice is greater tha the score o the first dice. A 6 = { (x, y) x < y } We shall list the possible sample poits of A 6, as A 6 = { (, ), (, ), (, 4), (, 5), (, 6), (, ), (, 4), (, 5), (, 6), (A 6 ) = 5 ad P(A 6 ) = (, 4), (, 5), (, 6), (4, 5), (4, 6), A 5 5 S 6 6 (5, 6) } (4) Four books o Physics, six books o Mathematics ad three books o Biology are to be arraged o a shelf. Fid the probability that books of the same subject are (i) together (ii) ever together. Solutio : The compositio of books o various subjects is as follows : Subject Physics Mathematics Biology Total No. of books 4 6 Table 7.5 Probability /

116 Thus total of books ca be arraged o a shelf i P =! ways. There are groups of books (books o oe subject forms a group) These groups ca be arraged i! ways. I each arragemet withi a group, 4 books o Physics ca be arraged i 4! ways, 6 books o Mathematics ca be arraged i 6! Ways, books o Biology ca be arraged i! ways. (i) (ii) Thus all the books i of the same subject ca be arraged i! 4! x6! x! ways. Let A be the evet that the books of the same subject are together. (A) =! 4! 6!! P(A) = = ! = = A S !! 6x4x6x6! 6x7x8x9xxxx The evet that books of same subject are ever together is the complemet of evet A.(ie A') P P( A ) P (A) = 9. Mathematics & Statistics / 4 A S (5) A box cotais 8 tickets bearig umbers,,, 5, 7, 8,. A ticket is draw at radom from the box ad kept aside, the the secod ticket is draw. Fid the probability that both tickets show odd umbers. Solutio : Two tickets without replacemet ca be draw i 8 C ways. ( S ) = 8 C. Two odd umbered tickets out 5 odd umbered tickets ca be draw i 5 C ways. 5c Probability = 8c (6) A purse cotais 4 silver cois ad 5 copper cois. Aother purse cotais silver ad 4 copper cois. A purse is selected at radom ad a coi is draw at radom. What is the probability that it is a copper coi? Solutio : 5 4 Purse Silver Copper Total A B 4 7 Table 7.6

117 (i) (ii) Probability of selectig a purse at radom is. There are two cases as : Purse A is selected ad a copper coi is selected from it. Its probability is 5 5 =. 9 8 Purse B is selected ad copper coi is draw from it. its probability is = Two cases are mutually exclusive. Hece the required probability is the sum of these probabilities. required probability = (7) A bag cotais 5 white ad 4 black balls ad aother cotais 4 white ad 6 black balls. Oe ball is trasferred from st bag to the d bag, ad a ball is draw from the d bag. Fid the probability that the ball draw is black. Solutio : 7 6 Bag White Black Total A B 4 6 Table 7.7 We shall cosider two cases depedig upo the colour of the ball beig trasferred. The compositio of bag chages accordigly. Case (i) : Let us suppose that a white ball is trasferred from bag A to a bag B. Probability of selectig a white ball from bag A is 9 5. Compositio of bag B ow becomes a white (5) ad black (6) : Total () Now probability of drawig a black ball from bag B is Hece the probability is =. 9 Case (ii) : Let us suppose that a black ball is trasferred from bag A to a bag B. Probability of selectig a black ball from bag A is 9 4. Compositio of bag B ow becomes : White (4), Black (7) : Total () 7 Next, probability of drawig a black ball from bag B is Hece the probability is = As the two cases are mutually exclusive, the required probability is the sum of probabilities of two cases. The required probability 8 58 = Probability / 5

118 (8) A room has 4 sockets for lamps. From a collectio of 5 bulbs of which 8 are defective, 4 are selected at radom ad put i the sockets. Fid the probability that the room is (i) dark (ii) lighted. Solutio : The compositio of bulbs is as (i) Good bulbs Bad bulbs Total Table 7.8 Of these 4 bulbs ca be selected i 5 C 4 ways ( S ) = 5 C 4. Let A be the evet that the room is dark. This is possible oly if all 4 bulbs selected are comig out of 8 bad bulbs. This ca be doe i 8 C 4 ways. (A ) = 8C 4 Hece P ( A, ) = A 8C S 5C 5 4 = 9 4 (ii) Let A be the evet that the room is lighted. This is complemet of the evet A P(A ) = l P(A l ) = l (9) A committee of 4 boys ad girls is to be formed from a group of 8 boys ad 5 girls selectig radomly. What is the probability that the committee cotais a particular boy ad a particular girl? Solutio : A committee of 7 members out of ca be formed i C 7 = C 6 ways. (S) = C 6. A committee of 4 boys ad girls cotaiig a particular boy ad a particular girl ca be formed as follows : boys out of 7 boys i 7 C ways ad girls out of 4 girls i 4 C ways. (A) = 7C 4C = 5 6. ways p(a) = A s C = () A card is draw from a pack of 5 playig cards. Fid the probability that the card draw is (i) red or bears a umber betwee 5 ad both iclusive (ii) a ace or a kig (iii) ace or spade (iv) diamod or a face card. Solutio : Sice a card is draw out of 5 playig cards. this ca be doe i 5 C = 5 ways. (i) (S) = 5. Let A be the evet that a card draw is red or bears a umber betwee 5 ad both iclusive. Mathematics & Statistics / 6

119 There are 6 red cards ad umber of cards bearig os. 5, 6, 7, 8, 9 are is 4. Out of these 4 cards, are red cards. ( A ) = = 8 Hece, P.(A ) = (A) 8 9. (S) 5 6 (ii) Let A be the evet that card draw is ace or kig. There are 4 aces ad 4 kigs. ( A ) = = 8 Hece, P(A ) = (A ) 8. (S) 5 (iii) Let A be the evet that card draw is ace or spade. There are 4 aces ad spade cards icludig a ace. ( A ) = 4 + = 6 p(a ) = (A ) 6 4. (S) 5 (iv) Let A 4 be the evet that card draw is a diamod or a face card. There are diamod ad face cards. Out of them, there are diamod face cards. ( A 4 ) = + = Hece, P (A 4 ) = 4 (A ) (S) 5 6 () A ur cotais 4 red, white ad black balls. If balls are draw at radom, fid the probability that they are of differet colour. Solutio : The compositio of the bag is as : Red (4) White () Black () Total () Table 7.9 Two balls ca be draw out of balls i C, = 9 = 45 ways. Let A be the evet that the balls draw are of differet colours. The followig are the possible cases of draw, ad the respective umber of ways i which this ca be doe. Case Red (4) White () Black () Total () (i) _ 4C C = (ii) _ 4C C = (iii) C C =9 Table 7. All these case are mutually exclusive. Hece (A) is the sum of these cases of selectio. ( A ) = = P(A) = (A). (S) 45 5 Probability / 7

120 Check your progress 7. () Fill i the blaks by choosig appropriate umber from the bracket. (i) A class cotais boys ad 6 girls. If a committee of is chose at radom from this class, the probability that exactly boys are selected is,,, (ii) Two me ad two wome are seated i a row at radom. The probability that wome are eighbours is..,,, 4 4 (iii) A cricket eleve is to be selected out of a group of 4 players. The probability that the team icludes at least oe of two specified players A ad B is..,,, (iv) Three cards are draw from a pack of 5 playig cards. The probability 7 7 that they cotai exactly two hearts is,,, (v) Two fair dice are rolled. The probability that the maximum of the two umbers is greater tha four is.,,, (vi) A ur cotais 5 red, white ad blue balls. If 5 balls are draw at radom the probability of obtaiig red, white ad blue ball is,,, 7 7 (vii) I a batch of electric bulbs, are defective. Two bulbs are selected at radom ad put ito sockets i a room. The probability that the room is illumiated is...,,, 7 7. ADDITION THEOREM ON PROBABILITY If A ad B are ay two evets the P(AB) = P(A) + P(B) P(AB) Proof : Fig. 7. Let A = { a a,.,a m, c, c, c k } B = { b b, b, c c. c k } Hece A B = { c, c, c k } ad AB = { a a,.a m, c, c c fc, b,, b b } P(A) = m i P (a k i ) p (cl ) l Mathematics & Statistics / 8

121 P(B) = j p (b k j ) P (Pl ) l P(AB) = P (c l ) k l Now P(AB) = m i P (a k i ) P(cl) P(b j) l j Cor. I. Cor. II m k = P (a i ) P (c l ) i l + j k k P (b j ) P (cl ) P (C l ) l l = P(A) + P(B) P(AB) i.e. P(AB) = P(A) + P(B) P(AB) If A ad B are mutually exclusive evets AB = ad P(A B) = P() = P(AB) = P(A)(B) If B = A' the AA'= ad AA' = S P(AA') = P(A) + P(A') P(AA) P(S) = P(A)+P(A') P() = P(A) + P(A') Thus P(A) = l P(A). We shall illustrate the use of the theorem i the followig examples : Example () The probability that a perso stoppig at a petrol pump will ask for petrol is.8 ad the probability that he will ask for water is.7 ad the probability that he will ask for both is.65. Fid the probability that a perso stoppig at this petrol pump ad will ask either petrol or water. Solutio : Let A be the evet that a perso asks for petrol ad B be the evet that a perso asks for water. we have P(A) =.8, P(B) =.7 ad P ( A B ) =.65 we have P ( A B ) = P(A) + P(B) P(AB) = =.85 i.e. a perso stoppig at a petrol pump will ask for petrol ad / or water has probability P(AB) =.85. The evet that he will either ask for petrol or for water is A B =(AB). P(AB) = l P(AB) =.85 =.5. Probability / 9

122 Example () If A ad B are two evets such that P (A ) =.8, P (B ) =.6 ad P (A B ) =.5, fid P(AB). Solutio : We have P(AB) = P(A) + P(B) P(AB) = =.9. Example () A ad B are two evets such that P (A B ) = 6 5, P(AB) = ad P(B) = ; fid P(A). Solutio : We have P (B ) = P ( B ) = Next P(AB) = P(A) + P(B) P(AB) P (A) = Check your progress 7. 5 = P(A) ) State whether true or false, if false correct the same ad write the correct aswer. (i) P(A) = 5 4,P(B) = 5 ad P (A B) = () P(AB)= 9 () P(AB)= () P(AB)= 4 () P(AB)= 5 (ii) A bowl cotais slips umbered to. A slip is draw at radom from the bowl. The probability that the slip bears a umber which is ( ) divisible by 5 is 5, 7 ( ) divisible by 7 is, 5 8 () divisible by 5 ad / or 7 is, 5 7 () divisible by 5 but ot by 7 is 5 (iii) A ur cotais white ad 5 red balls ad aother ur cotais white ad 4 red balls ad a ball is draw at radom. The probability that the ball is red is CONDITIONAL PROBABILITY Suppose that evet A has already happeed. The probability that evet B will happe is the coditioal probability of B give A ad is deoted by P (B / A) ad is read as; probability of B give A. Similarly probability that evet A will happe o the assumptio that evet B has already happeed is coditioal probability of A Mathematics & Statistics /

123 give B ad is deoted by P (A / B ) ad is read as : Probability of A give B. e.g. Let A be the evet that oe card draw from a pack is spade. It is kept aside ad the a secod card is draw. Let B be the evet that d card draw is also spade. P(A) = P (B/A) = ( spade cards are i a pack of 5 cards) 5 7 If the first card draw is ot, spade ad still a secod card draw is spade, P( B/A' ) = 5 We shall use Multiplicatio theorem as : P (A B) = P(A/B). P (B) or P(A B) = P(B/A). P(A) We shall verify it for the above illustratio. AB meas both cards draw are spade. (s) = 5 C ad (A B ) = C P (AB) = Hece P(B/A) = S A B P A A B 4 A B C S 5 C We shall illustrate this cocept by followig examples. Example () : A ad B are two, evets i a sample space such that P(A) =.6, P(B) =. ad P (A/B) =.5 Fid P [ A/( A B ) ] Solutio : We have P(AB) = P(A/B) P(B) Also by additio theorem, = (.5) (.) =. P(AB) = P(A)+P(B) P(AB) = =.7 Fig. 7. Next A (AB) = A as show i Fig. 7.. A / A B P = = P A A B P A B P PA.6 6 A B.7 7 Example () : I a certai examiatio out of 5 cadidates passed i Ecoomics, 5 passed i psychology ad failed i both the subjects. A Probability /

124 cadidate is selected at radom. Fid the probability that he has passed i Ecoomics, if it is kow that he has passed i Psychology. Solutio : Let E = Set of studets passed i Ecoomics. P = Set of studets passed i Psychology. Here ( S ) = 5, (E) =, (P) = 5. Fig. 7. Let x be the umber of studets passed i both subjects Ecoomics ad Psychology. The umber of studets passed i Ecoomics oly = x, ad umber of studets passed i Psychology oly is 5 x. (Refer to Fig. 7.) x + x + 5 x + = 5 (total umber of studets.) i.e. 75 x = 5 i.e. x = 5 = (EP) P (EP) = ad P(P) = E (P) (s) p 5 S 5 By coditioal probability, probability of studet selected at radom who has passed i Ecoomics, give that he has passed i Psychology is P (E/P) = P E P Mathematics & Statistics / P / 5 P 7 / INDEPENDENT EVENTS Two evets A ad B are said to idepedet if occurrece of oe of them does ot deped o the occurrece of aother. Thus evets A ad B are idepedet if P(A).P(B) = P(AB). We shall verify it for oe simple case : Let a sigle ubiased dice be rolled. Let A be the evet that score is eve umber; ad B be the evet that score is multiple of three. Here (S) = 6 We have A = {, 4, 6}, B = {, 6}, A B = {6} (A) =, (B) =, ( A B ) = Hece P(A) = P (B) = P (AB) = A S 6 B (S) A 6 B S 6

125 Thus we verify that, P (A). P (B) =. 6 = P (AB). Hece A ad B are idepedet evets Multiplicatio theorem : Whe two evets A ad B are idepedet the probabilities of occurrece of both the evets A ad B simultaeously is the product of the probabilities of evets A ad B. Thus whe A ad B are idepedet evets. P(AB) = P(A).P(B) I the above illustratio A B is the evet that the score is multiple of three ad a eve umber. Whe the evets are ot idepedet, the theorem is P(AB) = P(A).P(B/A) = P ( B ) P ( A/B ). We shall illustrate the above priciples by the followig examples. Example () : A fair dice is tossed twice. If A = the evet that sum of the two umbers is 7. B = the evet that the umber, i the secod toss is 6. Show that A ad B are idepedet. Solutio : (S) = 6, as ay of umber to 6 ca appear o top at ay toss. Here A = {(, 6), (6, ), (, 5), (5, ), (, 4), (4, )} (A) = 6 P (A) = (A) S 6 6 We have B = {(, 6), (, 6), (, 6), (4, 6), (5, 6), (6, 6)} (B) = 6 P (B) = 6 Also, AB = {(,6)} ( A B ) = P(AB) = A B S 6 6 ad P(A).P(B) =. P A B Thus P(AB) = P(A).P(B). Hece A ad B are idepedet evets. Example () : If A ad B are two evets such that P(A) = 4, ad P (B)= (A B ) =. Fid (i) P (A B (ii) P (B/A ) (iii) P (A/B ). Solutio : We have additio theorem P (AB) =P(A) + p (B) (AB) = = Probability /

126 (i) Thus P ( A B ) = (ii) P(B/A) = (iii) P(A/B) = B P A B P P A P 4. B B 4 Example () : If A ad B arc two evets such that P(A) =.7, P(B) =.4 ad P (A B) =. fid (i)p(ab') (ii) P (B/A ) (iii) P(A'B) Solutio : (i) By the adjoiig Ve diagram (Fig. 7.4) We have AE = A (AB) P(AB') = P(A) P(AB) =.7. =.4 P A B (ii) P (B/A) = P A (iii)..7 Fig By the Ve-diagram A'B=B (AB) P(A'B) = P(B) P(AB) =.4. =.. Example (4) : Give that P(A) = 5, P(B) =, P (B/A ) = fid (i) P (A/B ) (ii) P (A B) Solutio : We shall first fid out P(A B). By coditioal probability, we have P (B/A) = (i) P ( A/B ) = P (AB )/P (B ) = i.e. P (AB) = 4 /. 5 5 P A B P A P (A / B) (ii) P(AB)=P(A)+P(B) P(AB) = 5 5 Mathematics & Statistics / 4

127 P (AB) = Example (5) : I a group of equal umber of me ad wome, 8% me ad 54% wome are uemployed. Fid the chace that a perso selected at radom from this group is employed. Solutio : The compositio of employed ad uemployed from the group ad their respective probabilities are as. Prob. employed uemployed Me.9.8 Wome Table 7. The group cotais equal umber of me ad wome. Hece probability of selectig ma/woma is.46 =. The probability of selectig a employed ma from this group is (.9 ) = The probability of selectig a employed wome from this group is (.46 ) As these two cases are mutually exclusive the required probability of selectig a employed perso from the group is the sum =.69. Example (6) : A problem i Mathematics is give to three studets Rama, Govida ad Seeta whose chaces of solvig it are, ad respectively. 4 Fid the probability that the problem will be solved if they try idepedetly. Solutio : We shall first obtai the probability that the problem is ot solved. By puttig the data i a tabular form we get, Name pr. of solvig p pr. of ot solvig ( p) Rama Govida Seeta 4 Table 7. 4 As they try idepedetly, the probability that the problem will ot be solved is the product of the probabilities of ot solvig. Probability that the problem will ot be solved = The required probability of the problem beig solved is probability of the complemetary evet ad is Probability / 5

128 Note : Istead of cosiderig various cases like Rama solves, Govida ad Seeta does ot solve, it is coveiet to cosider complemetary evet. Example (7) : A husbad ad a wife appeared i a iterview for two vacacies i the office. The pr. of the selectio of the husbad is 7 ad that of wife's selectio is 5. Fid the probability that (a) both of them are selected (b) oe of them is selected (c) at least oe of them is beig selected. Solutio : The data ca be put i a tabular form as (a) husbad wife pr. of selectio Table 7. probability of both of them beig selected is, (b) Here we shall cosider two cases : Case (i) : pr. of rejectio Husbad is selected ad wife is rejected. The probability is, Case (ii) : Husbad is rejected while wife is selected. The probability is, (c) The two cases are mutually exclusive. The required probability is sum of them. The required probability = It is complemet of both beig rejected, probability that both are rejected is The required Probability = or It is sum of both (a) ad (b) above viz. either husbad is selected or wife is selected or both are selected. Probability is Example (8) : Three persos A, B ad C fire a target simultaeously. The probabilities that A, B ad C ca hit the target are, ad respectily. Fid the 4 5 probability that exactly two of them hit the target Mathematics & Statistics / 6

129 perso pr. of hittig pr. of ot hittig A B C We shall cosider three cases : 4 5 Table 7.4 (i) A, B hit ad C does ot hit (A, B ad C) (ii) B ad C hit, while A does ot hit (B, C ad A) (iii) A ad C hit, while B does ot hit (A, C ad B) The probabilities i each case are as, (i) 4 P(A).P(B).P(C) = (ii) P(B).P(C).P(A') = 4 5 (iii) P ('A ) P ( C ) P ( B') = 5 4 All these cases are mutually exclusive, the required probability of exactly two of them hittig the target is the sum of the probabilities. The required probability is = Check your progress () Fill i the blaks by choosig correct umber from the bracket. (i) Of girls i a class, have blue eyes. If two of the girls are chose at radom; the probability that () both have blue eyes is 4...,,, () either has blue eyes is 5 () at least oe has blue eyes is 7, 5 5 8, 5 7, 5 Probability / 7, 67,, 9 (ii) Three bolts ad three uts are put i a box. If two parts are chose at radom, the Probability that is a ut ad is a bolt is..,,, 5 5 () Give that P(A) = P(B) =, P ( A B ) =, Test for idepedece of A ad B. () The probability that a problem ca be solved by A, B, C is,, respectively. Fid the probability that the problem is ot solved THE PROBABILITY MODEL So far we have cosidered cases i which the probability of every sample poit is the same. I other words, we have cosidered equiprobable space. Now we shall cosider cases where the probability of each sample poit is differet. But the basic

130 coditios : P ( w i.) ad P(w i ) =, must hold here also. If ay of these coditios are ot fulfilled, it will ot be a probability model. e.g. () Let S = { x, x, x, x 4 } ad P (x ) = 4, P ( x ) =, P (x ) =, P ( x4 ) = 4. Here both the above coditios are fulfilled ad hece it is a probability model. () Let S = {a, b, c, d, e} ad P(a) = 5, p (b) = 5, P(c)= 5, P (d)= 5, P(e)= 5 6 Here, P(b) = ad su of the probabilities = Hece it is ot a probability model. Now we shall cosider examples where we have to fid out probabilities of various; sample poits from the data : Example () : Dice is loaded such that the occurrece of eve umber o top is twice as likely as the occurrece of odd umber. Fid the probability that the umber show o uppermost face is greater tha 4. Solutio : let x be the probability of occurrece of odd umbers. The x is the probability of occurrece of eve umber. The probability model is, umber pr. model x x x x x x actual pr Table Sum of the probabilities is 9x. Sice it is a probability model, 9x =. x = 9. Now we ca complete the above table 7.6. Let A be the evet that umber is greater tha 4. A = {5, 6} P(A) = P(5) + P(6) = 9 9 Example () : Three horses H, H ad H are i a race. H is twice as likely to wi as H ad H is twice as likely to wi as H. Fid their respective probabilities of wiig a race. Solutio : Let the probability that horse H wis a race be x. P(H ) = x Horse H is, twice as likely to wi as H P(H ) = P(H ) = x Further, H is twice as likely to wi as H. P(H ) = P(H ) = 4x As oly oe of them will wi the race, the sum of the probabilities is. Also these evets are mutually exclusive ad exhaustive. = P(H )+P(H ) + P(H ) i.e. = x + x + 4x = 7x Mathematics & Statistics / 8

131 i.e. x =. P(H ) = 7 4, P(H ) = P(H )= ad P(H ) = P(H )= Example () : There are three groups of childre. Group I cotais boys ad girl. Group II cotais boy ad girls ad Group III cotais boys ad girls; oe child is selected at radom from each group. Fid the probability that three selected childre comprise of girls ad boy. Solutio : The compositio of each group, possible modes of selectio have bee tabulated. The correspodig probabilities are expressed i brackets. Group Boys Girls Total Modes of selectio (probabilities) I B G G II G B G III 4 G G B Probability As the cases are mutually exclusive, the required probability = 9 Table Check your progress () Fill i the blaks by choosig appropriate umber from the bracket. (i) coi is weighted so that head is four times as likely to appear as tail. 4 P(H) =...,,, (ii) A dice is loaded such that appearig 4 o a top is certai. P(4) =...,,, 6 () State whether each of the followig is a probability model o S = {S, S, S, S 4, S 5 } (i) P(S ) =,P(S ) = 5,P(S )= 5,P(S4 ) = 5 P(S5 )= 5 (ii) P(S ) = 6.P(S ) =,P(S )= 4,P(S4 ) = 5,P(S5) (iii) P ( S ) = P ( S ) =, P ( S ) =, P ( S4 ) =, P ( S5 ) = Probability / 9

132 7.7 SUMMARY Probability is a theory of chace. I each ad every subject there is probability of pass ad fail we wat to be half ad half. Whe we have to take some decisio the probability of positive activity should be more tha half. We lear this cocept with games like tossig a coi, cois dice, dices, playig cards draw radomly. But importat is equal to oe ad ot more tha oe. 7.8 CHECK YOUR PROGRESS - ANSWERS 7. 7 () (i) 56 (ii) 4 55 (iii) 9 7 (iv) 85 (v) (vi) (vii) 7. () (i) () true () False, correct is () true () false, correct is (ii) (iii) () true () true 8 9 () false, correct is () false, correct is 5 5 true 7.4 ad 7.5 () (i) () 5 () 5 7 () 5 8 (ii) 5 () idepedet () () (i) 5 4 (ii) () (i) yes (ii) o (iii) o 7.9 QUESTIONS FOR SELF - STUDY ) Two dice are rolled. Write dow the sample space for the experimet. Hece write dow the followig evet sets a) The umbers o both dice are idetical. b) The sum of umbers appearig o them is divisible by 4. ) Three cois are tossed. Fid the probability that at least two heads appear. ) A room has electric lamps. From a collectio of 5 electric bulbs of which oly are good are selected at radom ad put i the lamps. Fid the probability that the room is lighted by at least oe of the bulbs. Mathematics & Statistics /

133 4) The probability that A ca shoot at a target is 5/7 ad the probability that B ca shoot at the same target is /5 A ad B shot idepedets. Fid the probability that a) the target is ot shot at all b) the target is shot at least oe of them 7. SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby. Statistical Aalysis: A Computer - Orieted Approach Itroductio to Mathematical Statistics by S. P. Aze & A. A.Afifi Probability /

134 NOTES Mathematics & Statistics /

135 CHAPTER 8 BINARY SYSTEM 8. Objectives 8. Itroductio 8. DECIMAL system 8.. Coversio of a umber i DECIMAL system 8. BINARY system 8.. Coversio of a umber i BINARY 8.4 OCTAL umber system 8.4. Coversio of a umber to OCTAL 8.5 HEXADECIMAL umber system 8.5. Coversio of a umber to HEXADECIMAL 8.6 Coversio of DECIMAL TO BINARY 8.6. Coversio of BINARY to DECIMAL 8.7 BINARY ARITHMETIC 8.7. ADDITION 8.7. SUBTRACTION 8.8 Summary 8.9 Check your Progress - Aswers 8. Questios for Self - Study 8. Suggested Readigs 8. BJECTIVES After studyig the cocept of a BINARY SYSTEM you will be able to use ad solve problems related to the followigs: ) BINARY NUMBERS Coversio of BINARY to a) DECIMAL b) OCTAL c) HEXADECIMAL ) BINARY ARITHMATIC : a) ADDITION b) SUBTRACTION 8. INDRODUCTION There are differet umber systems we leart , are te atural umbers, o is added to set of these umbers.so we get total te umbers to study mathematics. There are four basic operatios ADDITION, SUBTRACTION, MULTIFICATION & DIVISION.I DECIMAL SYSTEM te digits are used as,,,,4,5,6,7,8,9. But computer do ot uderstad these umbers. Every computer stores umbers, letters & other special characters i coded form. 8. DECIMAL SYSTEM = = = = = 4 Biary System /

136 8.. Coversio of a umber ito DECIMAL SYSTEM: PLACE i NUMBERS: TL LAKHS TT THOU HUNDERS TENS UNIT PLACE VALUES: PLACE NAME PLACE DIGIT PLACE VALUE VALUE UNIT 4 4* 4 TENS 9 9* 9 HANDERD 8 8* 8 THOUSAND 7 7* 7 TENTHOUSAND 6 6* 4 6 LAKH 5 5* 5 5 TENLAKHS * 6 CRORES * = * 7 + * 6 + 5* * * + 8 * +9* + 4* 8. Check your progress: Express the give umbers i DECIMAL NUMBER system: ) 567 ) 956 ) 4) 56 5) 9 8. BINARY SYSTEM Computer uderstads oly two digits, that meas oly two umbers. BI meas two so this umber system is called BINARY NUMBER SYSTEM.The BASE of Biary system is. 8.. CONVERTIN umber to BINARY SYSTEM: EX: 5 5 REMENDER = EX: REMENDER 5 = Mathematics & Statistics / 4

137 8.4 OCTAL NUMBER SYSTEM OCTAL meas 8.Base of this umber system is 8,that is it use oly eight digits from the te umbers of DECIMAL SYSTEM So umber of digits are 8 but the last umber is : Coversio of umber ito OCTAL umber system: EX: REMENDER = HEXADECIMAL NUMBER SYSTEM HEXA + decimal meas it is a umber system usig 6 digits to write a umber. Digits the te digits ad more six digits from capital letters from Eglish laguage A,B,C,D,E,F. Where A is th B is th C is th D is 4 th E is 5 th F is 6 th So largest digit is F or 5 which is oe less tha the base 6. Each positio i HEXADECIMAL system represets a power of base Coversio of a umber ito the HEXADECIMAL system: EX: Remiders i HEXADECIMAL system =F 6 =A = 4 = AF 6 EX: 8.6 CONVERSION OF DECIMAL TO BINARY REMENDER = 8.6. COVERSION OF BINARY to DECIMAL EX: FIVE digested umber is give So maximum power of is four Biary System / 5

138 = * 4 + * + * + * + * EX: 95 = = 5 CONVERSION OF DECIMAL TO OCTAL 8 95 Remaider = BINARY ARITHMATIC Basic arithmetic operatios are ADDITION (+).SUBTRACTON (-).MULTIPLICATION (*).DIVISION (/). These operatios are iterrelated.we use that i previous staders. But i BINARY system oly two digits are used so there are some RULES to calculatios 8.7. RULES for ADDITION: ) + = ) += ) += 4) += PLUS CARRT OF to ext higher colum is the largest digit i biary umber system ay sum greater tha requires a digit to be carried over. EX: ) + This is the additio DECIMAL SYSTEM.But i BINARY system BINARY SUBTRACTION: IN subtractio we have to borrow a umber from left digit this umber is depedig o the base i which we are subtractig. EX: 5- RULES: ) -= ) -= WITH BORROW from the ext colum ) -= 4) - = DECIMALSYSTEM BINARY SYSTEM =* + * Mathematics & Statistics / 6

139 8.8 SUMMARY Biary system is very importat for the computer studets. Computer ca uderstad oly cocept of ad meas yes / o. For ay mathematical umber there is proper arragemet of ad. There are also other systems e.g. Ocatal, Hexadecimal. The base for each system is differet CHECK YOUR PROGRESS - ANSWERS. 567 = x + 5 x + 6 x + 7 x. 956 = 9 x 4 + x + x + 5 x + 6 x. = x + x + x + x = 5 x + 6 x + x + x 5. 9 = 9 x + x + x + x 8. QUESTIONS FOR SELF - STUDY Add biary umbers i both biary ad DECIMAL forms..,.,.,., 5., SUBTRACT the followig umbers i BINARY & DECIMAL both the system: ) 5 5 ) ) 6-7 4) - 5) - 9 6) - 7) - 8) - 9) - ) - 8. SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Discrete Mathematical Structures for Computer Sciece by Berard Kolma ad Robert C Busby. Mathematics ad Statistics by M. L. Vaidya ad M. K. Kelkar Biary System / 7

140 NOTES Mathematics & Statistics / 8

141 CHAPTER 9 MATHEMATICAL LOGIC AND TRUTH TABLE 9. Objectives 9. Itroductio 9. Statemet OR Propositio 9.. Defiitio 9.. Truth Value of Statemet 9. Use of Ve diagram 9.4 Logical Coectives 9.5 Diagrammatic Represetatio of Logical coectives 9.5. Negatio (NOT) 9.5. Cojuctio (AND) 9.5. Disjuctio 9.6 Truth Table 9.7 Tautology 9.8 Cotradictio (Fallacy) 9.9 Summary 9. Check your Progress - Aswers 9. Questio for Self Study 9. Suggested Readigs 9. OBJECTIVES Dear Frieds this is ew mathematical additio to your laguage. After studyig this chapter you will be able to Thik Logically Differetiate the logic ad mathematical logic Explai Bits Explai Yes or No 9. INTRODUCTION Mathematics is a way of thikig ad reasoig. Logic is the disciplie that deals with art of reasoig. Systematic reasoig is a base of Mathematics. Study of correct ad systematic reasoig.everybody thiks but everybody caot distiguish betwee good ad bad thiks.logic gives idea to show how oe should thik if oe has to thik clearly. Logic was first give by George Boole, so Mathematics logic is called as BOOLEAN LOGIC. Symbolic logic is very importat i Computers. 9.. Defiitio 9. STATEMENT OR PROPOSITION A statemet or propositio is a declarative setece which is either TRUE or FALSE but ot both. EX: ) is a eve prime umber. ) Three plus two is equal to six. ) Mumbai is capital of Idia. Mathematical Logic ad Truth Table / 9

142 I Logic the above seteces are true or false, but ot both (true ad false). This is called Law of excluded middle EX: ) Work hard ) College life is very good. ) Ope the classroom. We deote statemet by small letters, p, q, r, s.etc. EX: ) p= Square of a umber is odd umber. ) q= This is Mathematics classroom ) r= study hard for your exam. 9.. Truth Value of a Statemet Defiitio: Truth value If a statemet is true the its truth value is defied as to be T (or ) ad if statemet is false its truth value is F (or ). EX: 4 ) p= Square of a umber is odd. ANS: Statemet is false. Truth value = F Ex: 5 q: This is Mathematic s classroom. As: Statemet is true. Truth value =T Ex:6 ) r = study hard for your exam. Statemet is true. Truth value = T Check your Progress 9. Write truth values of followig statemets ) + 5 = ) Pue is ot a big city. ) Zero is a atural umber. 4) is a ratioal umber. 5) Empty set is a subset of every set. Mathematics & Statistics / 4

143 9.. All Xs are Ys- 9. USE OF VENN DIAGRAM The statemet is true with true value T. X is proper subset of Y. X Y Ex: 7 ) All Natural Number are whole umbers. ) All eve umbers are atural umbers. 9.. Type )-NO Xs are Ys U Ex: 8 ) X is set of eve umber s. ) Y is set of odd umbers. ) U is set of atural umbers. 9.. Type 4) Some Xs are Ys XY 9..: Type 4-ALL Ys are Ys All Xs are Xs - X =Y Mathematical Logic ad Truth Table / 4

144 ) NOT ) AND ) OR 4) IF.THEN 5) IF AND ONLY IF 9.4. Simple statemet LOGICAL CONNECTIVES A combiatio which does t iclude ay logical coective is called a simple statemet. EX: ) Today is Suday. ) Newto was my fried Compoud statemet A combiatio of simple statemet formed by usig logical coectives is called compoud statemet. EX:9 ) Seeta ad Geeta dacig ) x= or y= Negatio of a statemet (NOT) Defiitio - If P is ay statemet the NOT P is called the egatio of the statemet P ad is deoted by ~ p. EX: ) is a prime umber. ) Three plus two is five. P: is a prime umber. ~ p : is ot a prime umber. ) q: Three pulse two is five. ~ q: Three plus two is ot five Cojuctio (AND) (p Λ q) If p ad q are two simple statemet.the the compoud statemet p ad q is called there cojuctio. It is deoted by pλq Read as p ad q ) p: 5 is a odd umber. ) p: It is soft. q: 5 is a perfect square. P Λ q: 5 is a odd umber ad perfect square. q: It is good. p Λ q : It is soft ad good Disjuctio (OR Alteratio) (pvq) EX: Defiitio: If p, q are two simple statemets the compoud statemet. It is deoted by Read as p: I will study at home. p V q p or q q: I will go to my fried s house for study. p V q: I will study at home or go to my fried s house. Mathematics & Statistics / 4

145 EX: Here p: xє A q: xє B p V q: x Є A or x Є B p or q meas - either p or q Or both p ad q X Є A meas x Є A or x Є B OR X Є A x Є A ad x Є B Implicatio or Coditioal statemet (p q) Defiitio: If p ad q are two simple statemets the compoud statemet if p ad q is called a coditioal or a implicatio. It is deoted by Read as p q p implies q EX: ) p: is eve umber. ) q: is divisible by. P q; If is eve umber the it is divisible by. ) p: Lies L ad L are parallel lies. P q: Lies LadL have o itersectio poit. q: If lies L ad L are parallel lies the they have o itersectio poit : Double implicatio or Bicoditioal (p q) Defiitio: If p ad q are two statemets, the the compoud statemet p if ad oly if q ( or p if q ) is called bicoditioal or double implicatio or equivalece. It is deoted by Read as p q p if ad oly if q EX; ) p: Triagle ABC has three sides. q: Triagle ABC is a triagle if ad oly if it have three sides ) p: Pratik studies a Egieerig. P q: Pratik passed th sciece examiatio with 9%marks. q Pratik studies a Egieerig if ad oly if he passed th sciece with 9% marks. The various coectives TABLE: Coectives Symbol Compoud Statemet NOT ~ Negatio OR V Disjuctio AND Λ Cojuctio IF.THEN IF AND ONLY IF Implicatio Double Implicatio 9.5 DIAGRAMMATIC REPRESENTATION OF LOGICAL CONNECTIVES 9.5.: NEGETION (NOT): EX: 5 ) p: x is a iteger. ~ P: x is ot a iteger. Mathematical Logic ad Truth Table / 4

146 ) P: A is set of umbers from to. ~ P: A is set of umbers other tha to. U A A Note: The Ve diagram for the egatio is similar to that of complemeted of a set. 9.5.: CONJUNCTION (AND) EX: 6 p: Ram is itelliget. q : Ram always studs first i the class. pλq: Ram is itelliget ad always studs first i the class. pλq 9.5.: DISJUNCTOIN: (OR) EX: 7 p: + =5 q: 9- =5 PVq: + = 5 or 9 - = 5 Mathematics & Statistics / 44

147 PVq 9.5.4: Implicatio: (IF.THEN): P: I go mad. q: I bite you. p q : If I go mad the I bite you : Double Implicatio (IF AND ONLY IF): P: Triagle ABC is isosceles triagle. q : Base agles of triagle ABC are cogruet. P q; Triagle ABC are isosceles if ad oly if its base agles are cogruet. 9.6 TRUTH TABLES 9.6.: Truth table for NEGATION (~P): Rule: If p has truth value T Mathematical Logic ad Truth Table / 45

148 The ~p has truth value F Ad if p has truth value F The ~p has truth value T Thus statemet ad its egatio have always opposite truth value s. P ~p T F F T EX: 8 p: 4 is a prime umber. ~ P: 4 is ot a prime umber. P ~P T F F T 9.6.: TRUTH TABLE for DISJUNTION (OR) RULE: If p ad q true Otherwise P ad q false pvq is false. pvq is true. pvq is false. P Q PVq T T T T F T F T T F F F 9.6. TUTH TABLE for CONJUCTION ( AND ): Rule: Both p ad q TRUE the pλq TRUE. Otherwise pλq FALSE. P q PΛq T T T T F F F T F F F F 9.6.4: If p THEN q CONDITIONAL Rule: Both p ad q TRUE The p-q TRUE. If p is TRUE, q is FALSE the p-q FALSE. If p is false, q is true the p-q TRUE. If p is false. q is false the p-q TRUE. P q P q T T T T F F F T T F F T EX: 9 p: Today Is Suday. q: It is holiday. Mathematics & Statistics / 46

149 P q : If today is Suday the it is holiday. P q P q T T T T F F F T T F F F TRUTH TALE for IF AND ONLY IF (BICONDITONAL): Rule: p ad q TRUE, the p q TRUE. P ad q FALSE, the p q FALSE. P or q FALSE p-q FALSE. P q P q T T T T F F F T F F F F Check your progress 9.6 Write TRUTH TABLE of the followig: ) pv ~p ) ~p Λq ) ( p Vq) V ~p 4) ~( p Λ ~ q ) 5) (p Λ q ) - ~(~p V ~ q ) ) 9.7 TAUTOLOGY Defiitio: A statemet patter which is true (always takes value oly TRUE) EX: p V ~ p TRUTH table P ~ p P V ~ p T F T F T T ANS: It is tautology. Sice the TRUTH table values of a tautology is always TRUE,the last colum of the TRUTH TABLE of tautology has all Ts oly EX:[pΛ ( p q ) ] q P q p q P Λ ( p q ) [pλ ( p q ) ] q T T T T T T F F F F F T F F F F F T F T ANS: It is Tautology. 9.8 CONTRADICTION (Fallacy) Defiitio: Statemet which is always false for all TRUTH values is called a CONTRADICTION. EX: p Λ ~ p Mathematical Logic ad Truth Table / 47

150 P ~ p P Λ ~ p T F F F T F Last colum of TRUTH table of ( p Λ ~p )is F. It is CONTRADICTION (Fallacy). Check your progress 9.8 Usig TRUTH tables check which of the followig statemet is a TAUTOLOGY or CONTRADICTION or either ) p V q ) ( p V q ) - q ) ( p Λ q)λ ~ q 4) ( pv q ) Λ ~ q 5) ( pλ q ) Λ ~ q 9.8 SUMMARY Mathematical Logic meas we take a proper meaig of Mathematics. If proper coditio ad situatio is give we ca trasfer it i mathematical relatio as variable x ad double of it meas x. Tautology meas all seteces are true with give proper coditio. Cotradictio is, we cotradict the situatio, the coditio give i the setece. e.g. is prime umber. Setece is true. But is divisible by, this cotradict the coditio of prime umber. 9.. False. False. False 4. False 5. True 9.5. p Λ ~q 9.9 CHECK YOUR PROGRESS ANSWERS p ~p ~ p Λ ~ q T F T F T T. ~ p Λ~q p ~p q ~ p Λ q T F T F F T F F F T T T T F F F Mathematics & Statistics / 48

151 . (p V q) V~ p p q p Λ q ~p (p V q) V~ p T T T F F T F T F F F T T T T F F F T F 4. ~ (p Λ ~ q) p q ~ q (p Λ ~ q) ~ (p Λ ~ q) T T F F T T F T T F F T F F T F F T F T 5. (~p V ~ q) p q p Λ q ~ p ~ q (~p V ~ q) T T T F F F T F F F T F F T F T F F F F F T T T p ~p p V q T T T As. : Not Tautology T F F Not Cotradictio F T F F F F p q p V q ~ q (p V q ) V ~q T T T F F As. : Not Tautology T F F T T Not Cotradictio F F F T F F T F F F T F F T F. Cotradictio 4. Tautology 5. Not cotradictio, Not Tautology Mathematical Logic ad Truth Table / 49

152 9. QUESTIONS FOR SELF - STUDY Solve the followig problems.. Give statemets p : Pratik is good studet q : Pratik is hoest studet Write i) ~ p ii) ~ q iii) p Λ q iv) p V q v) Truth table of p Λ q ad p V q. Draw Ve diagram of p : is positive umber q : is eve umber i) p ii) q iii) ~ p iv) ~ q v) p Λ q vi) p V q. Check for Tautology p : is odd umber q : 5 is divisible by oe ad itself Write ii) ~ p iii) ~ q iv) p Λ ~q v) ~ p V~ q 9. SUGGESTED READINGS. Pre-degree Mathematics by Vaze, Gosavi. Mathematics ad Statistics by M. L. Vaidya ad M. K. Kelkar. Statistical Aalysis: A Computer - Orieted Approach Itroductio to Mathematical Statistics by S. P. Aze & A. A.Afifi Mathematics & Statistics / 5

153 NOTES Mathematical Logic ad Truth Table / 5

154 NOTES Mathematics & Statistics / 5

155 Part -II Statistics

156

157 CHAPTER Itroductio to Statistics. Objectives. Itroductio. Defiitios of Statistics. Importace of Statistics.4 Scope of Statistics.5 Summary.6 Check Your Progress - Aswers.7 Questios for Self Study.8 Suggested Readigs. OBJECTIVES I our everyday life we make use of umbers or figures. These umbers is a iformatio expressed i umerical form ad is geerally refered as data i statistics. It may also be i the form of tables. After studyig this chapter you will be able to explai termiologies such as statistics, statistical methods. discuss ad explai the eed of ad scope of statistics i differet fields.. INTRODUCTION The word statistics seems to be derived from the word statist, the kow use of which dates back to 6, whe it was used i Hamlet by Shakespeare. The umerical iformatio used by the statists for the purpose of admiistratio of state was termed as statistics. At preset the word statistics is used to mea umerical data pertaiig to some departmet of iquiry ad it also meas the sciece of Statistics which icludes a umber of statistical methods such as collectio, classificatio, aalysis ad iterpretatio of umerical data.. DEFINITIONS OF STATISTICS Differet persos have defaced statistics i differet ways. Some of them have defied statistics as umerical data ad the others have defied it as a sciece. Most of them have described statistics as it appealed to them. Therefore oe of the defiitios has defied statistics quite comprehesively. Some of these defiitios are give below. a) Webster defied statistics as The classified facts represetig the coditio of people i a state, especially those facts which ca be stated i umbers or i tables or i ay tubular or classified arragemet. This defiitio is too arrow as it cofies the scope of statistics oly to such facts ad figures that represet the coditio of the people i a state. Sice statistics may represet various other facts such as biological, physical, commercial ad others, this defiitio is quite iadequate. b) Prof. Horace Secrist has give a more comprehesive defiitio which read as statistics are the aggregates of facts affected to a marked extet by multiplicity of causes, umerically expressed, eumerated estimated accordig to reasoable stadards of accuracy, collected i a systematic maer for a predetermied purpose ad placed i relatio to each other. This defiitio clearly poits out the characteristics which umerical data must possess i order that they ca be called statistics. Itroductio to Statistics / 5

158 Check Your Progress.-.. What is statistics?. How statistics id defied by Webster?. IMPORTANCE OF STATISTICS Importace of Statistics as a sciece lies i the service it has redered to the makid. I recet years, the growth of statistics has made itself felt i almost every phase of huma activity. It o loger cosists merely of collectio of data ad presetig it i tables or diagrams. It is ow cosidered to ecompass the sciece of makig decisios i the face of ucertaity. This covers cosiderable groud sice ucertaities are met whe we roll a die, a doctor treats a patiet, a actuary determies life isurace premiums, meteorologists make weather forecasts, a broker makes predictios about prices of shares, a ewspaper predicts a electio, so o ad so forth. Statistics, i its preset state of developmet, ca hadle most of the situatios ivolvig ucertaities. It at least provides the models that are eeded to study situatios ivolvig ucertaities. Statemets, i ay form become stroger, precise ad more appealig whe they are supported by relevat statistics. Statistical methods provide tools to summaries the complex umerical data ad to preset them i a maer which is easily itelligible. Statistics has provided techiques like statistical quality cotrol which have made revolutioary chages i the field of idustrial productio. Statistical methods are widely used i the field of agriculture i estimatig yield of a crop, i testig effectiveess of fertilizers, methods of irrigatio ad water maagemet, i developig ew varieties of seeds etc. To sum up oe ca say that it is hardly possible to sigle out a departmet of huma activity where statistics has ot creeped i. It has rather become idispesible i all phases of huma edeavour..4 SCOPE OF STATISTICS The field of applicatio of statistics is expadig very fast i moder times. It is of immese value ot oly to the admiistrators of a state but also ecoomists, busiessme, scietists ad research workers i sociology ad psychology as well. I idustry, statistical methods are used i estimatig demad for a productio i future ad estimatig the eed for a raw material, labour, fiaces etc. I large scale productio the statistical quality cotrol techiques are used to reduce rejectio of the product ad wastage of raw materials, which results i icreasig profits by reducig cost of productio. Statistical techiques are widely used i Ecoomics also. Ecoomics is maily cocered with the productio ad distributio of wealth ad with the cosumptio, savig ad ivestmet of icome. Statistics is also used i formulatig taxatio policies. The ecoomists have to deped o statistics to a great extet i solvig problems cofroted i productio ad distributio of essetial commodities. The policies of reducig uemploymet, poverty, risig prices etc. also deped o Statistics to a fairly good extet. The maagemet techiques are developig very fast i the twetieth cetury due to eormous growth of idustry ad busiess. Decisio makig is the prime fuctio of ay maagemet ad the statistical iformatio ad statistical techiques provide a soud basis for all sorts of decisio. Sice the complexities of busiess eviromet make the process of decisio makig difficult, the decisio maker caot Mathematics & Statistics / 54

159 rely etirely upo his observatio, experiece or evaluatio to make a decisio. Decisios have to be based o data which show relatioship, idicate treds ad show rates of chage i various relevat variables. Statistics provides methods for collectig, presetig, aalysig ad iterpretig meaigfully such data which is helpful i better decisio makig. The various statistical tools guide a maager i selectig the best course of actio uder give circumstaces. The decisios relatig to productio, pricig purchasig ad cotrollig various activities are redered easier with the help of statistics. There is a wide scope for applicatio of statistical methods i the field of basic scieces like Biology, Astroomy, Meteorology, Physics, Chemistry etc. Research work carried out i differet braches of sciece proves that it is impossible to coduct ay research without the help of statistics. It is used as a scietific method i developmet of differet braches of sciece. Statistical methods are used i establishig laws ad priciples i sciece ad also i validatig the same. I the field of medical sciece almost all the coclusios are based o observatios ad experimetatio. The statemets like 'smokig is ijurious to health, 'chewig tobacco causes cacer' are the statistical coclusios based o systematically collected data. Statistical methods are used i plaig the experimets ad aalysig the result for testig the effectiveess of differet medicies ad their hazards. There ca be o research i the medical scieces i moder times, without beig supported by statistics. The usefuless of statistics as a scietific method of studies i sociology ad psychology has bee widely recogised i moder age. The sociological studies are based o properly desiged sample iquiries which ivolve plaig of iquiry, collectio of data, aalysis ad iterpretatio of these data. Sice these scieces are ot exact scieces, the observed facts ca be hadled more purposefully oly by usig statistical techiques. These studies are useful for plaig ad executio of social welfare activities to be udertake by govermet or private agecies. Check Your Progress. -. &.4. List the use of statistical methods i agriculture field.. List the use of statistical methods i medical sciece field.. List the braches of basic sciece i which statistical methods are used..5 SUMMARY This chapter explais i detail the importace of statistics to makid ad scope of statistics i differet fields. Thus we see that statistics is very importat subject ad is useful i almost all areas.. &..6 CHECK YOUR PROGRESS - ANSWERS. The umerical iformatio used by the statists for the purpose of admiistratio of state was termed as statistics. Itroductio to Statistics / 55

160 . Webster defied statistics as 'The classified facts represetig the coditio of people i a state, especially those facts which ca be stated i umbers or i tables or i ay tubular or classified arragemet.'. &.4. Statistical methods are widely used i the field of agriculture i estimatig yield of a crop, i testig effectiveess of fertilizers, methods of irrigatio ad water maagemet, i developig ew varieties of seeds etc.. I medical sciece field Statistical methods are used i plaig the experimets ad aalysig the result for testig the effectiveess of differet medicies ad their hazards. Also for research.. Applicatio of statistical methods i the field of basic scieces like Biology, Astroomy, Meteorology, Physics, Chemistry etc..7 QUESTIONS FOR SELF - STUDY. Explai the differet meaig of the word Statistics. Give the defiitios of statistics by Webster, Secrist. What is the role of statistics i ecoomic plaig? 4. Describe the scope of statistics i busiess ad idustry. 5. Explai the role of statistics i social scieces. 6. Describe the importace of statistics i maagemet sciece..8 SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi Mathematics & Statistics / 56

161 NOTES Itroductio to Statistics / 57

162 NOTES Mathematics & Statistics / 58

163 Chapter Statistical Data. Objectives. Itroductio. Nature of Subject. Laguage of Statistics.. Populatio.. Variables.. Size of Populatio..4 Discrete ad Cotiuous Variables.4 Classificatio of data.4. Classificatio by attributes.4. Classificatio of variables.5 Graphical represetatio of data.5. Histogram.5. Frequecy polygo.5. Ogive curves.6 Diagramatic represetatio of data.6. Simple bar diagram.6. Subdivided bar diagram.6. Pie diagram.7 Summary.8 Check your Progress - Aswers.9 Questios for Self - Study. Suggested Readigs. OBJECTIVES After studyig this chapter you will be able to :- explai terms - attributes, variables, raw data, classificatio of data, populatio sample draw graphical represetatio of data (Histogram Frequecy polygo ad Ogive curves) describe Diagramatic represetatio of data. (Simple bar diagram subdivided Bar diagram, Pie diagram). INTROUDCTION Statistics i cocered with scietific methods for collectig orgaisig, summerisig, presetig ad aalysig data as well as drawig coclusios ad makig reasoable decisios o the basis of such aalysis.. NATURE OF SUBJECT Suppose we wat to compare the performace i Mathematics of two divisios i the examiatio. The first thig we have to do is to collect the marks of studets. These marks are collected is called, Data Hece first step of statistics is to collect the data. But, merely lookig at the mark lists, we will ot get ay idea about performace of studets i two divisios uder cosideratio. We have to fid, i) umber of fail studets Statistical Data / 59

164 ii) i.e. studets gettig less tha 4 marks. umber of pass studets, i.e. studets gettig equal to ad more tha 4 marks. Here agai we have cosider the class of studets meas I class Marks 6 & up to 74 above II class 4 to 59 Distictio 75 ad above This all meas hat we must make Classificatio, of the collected data. Hece, broadly speakig we ca say that, ) Collectio of data ) Classificatio of data ) Diagrammatic represetatio of data 4) Aalysis 5) Iferece are the differet aspects of this subject (statistics). Samplig : Whe a populatio is very large or ifiite practically it is ot possible to collect desired iformatio o all the uits of populatio. This may happe eve i case of small or fiite populatio whe measuremets of a variable is costly or i some cases destructive i ature. I such cases we select a small group of uits draw from the populatio to carry out ivestigatio. This small group of uits draw from the populatio is called a sample. e.g. to fid average height of studet studyig i a college istead of takig measuremet of height o all studets of college (populatio) we select a small group of o. of studets (sample) ad take measuremets o such selected studets. Similarly to estimate the average life of electric bulbs produced by a compay a sample of bulbs is take from the large populatio of o. of bulbs produced ad the life time of each bulb from a sample is determied by actually burig out the bulbs. The o. of uits i the sample is called the size of that sample. I real life there are may situatios where we use sample from populatio for makig judgemet about the populatio. Followig are few examples () For judgig the quality of rice i a bag we pick up hadful of rice ad judge the quality of rice i bag. () The average yield of crop ca be estimated by selectig a sample of farms ad fidig the mea yield per hectare for these forms. () A housewife cofirms whether the food is properly cooked or ot with the help of few particles take out of the cotaier. Clearly the food i cotaier is populatio ad food take out of cotaier for ispectio is sample. (4) For testig quality of milk a small quatity of milk is tested istead of etire bulk. Cocept of populatio ad sample ca be easily uderstad from followig diagram Mathematics & Statistics / 6

165 . LANGUATE OF STATISTICS Every subject has got its ow special termiology; the special words are used for special purpose. So the terms used i, are.. Populatio I commo laguage used word Populatio as o of people live i that particular area. But i statistics, we use this word, Populatio for ay Collectio of articles (items) uder cosideratio of our study purpose. e.g.- ) Studets i class ) Workers i Idustry ) Radio sets 4) T. V. sets 5) Variety of Mobiles available ow a days. 6) Agricultural field yields. So each member, object or observatios of the populatio is called, a idividual or member or elemet of that populatio. The populatio is also called a Uiverse... Variables Each idividual i the populatio is studied for a certai character or characteristic. e.g. Height Weight Marks i a subject Rai fall i a regio Yield of productio of variety of crop Productio i factory. Variables are of two types I) Quatitative ca be couted as a umber. e. g. Height ad weight of a studet. Temperature recorded i Moth of May II) Quatitative Whe the character is qualitative i ature ad hece ot expressible i umerical forms, it is called qualitative (a attribute) e. g. Sex, Religio. Mother togue. Faculty Natioality. Statistical Data / 6

166 Noe of these ca be expressed umerically, but each will divide a populatio is two or more groups; as Sex gives you o. of males & o, of females. Mother togue Marathi, Hidi, Tamil, Guajarati... Size of Populatio The umber of uits costitutig the populatio is called size of that populatio. e. g. B. C. A. = Studets. Size of Populatio = So this populatio is called, Fiite Populatio. The other is I fiite Populatio. e.g. N = Set of Natural Numbers = {,,.} Number of elemets are ifiite meas ucoutable...4 Discrete ad Cotiuous Variables I) Discrete variable A variable is said to be discrete if it a takes distict ad isolated values. e. g. Number of daily accidets i city. Number of family members. Number of decayed teeth of a child. II) Cotiuous Variable A Variable is said to be cotiuous whe it takes all possible values i a iterval. e.g. weights of persos i a group Temperature of Certai place.4 CLASSIFICATION OF DATA The raw data are very difficult to uderstad ad we caot draw ay coclusios from them uless we process it. The data so obtaied after processig is called as secodary data. Suppose a collectio of o a certai characteristics. Such a set of umbers does ot help i drawig ay coclusio about the data. The data ca be made more meaigful by a ordered arragemet or by dividig it ito differet groups or classes. This process is called, Classificatio of Data..4. Classificatio by attributes Whe the characteristics uder cosideratio is qualitative types or a attribute; the simplest way of classificatio is to put all the items or uits possessig that attribute i oe class ad remaiig items i other class. Such classificatio is called simple classificatio by attribute or dichotomy. e.g. we may classify group of persos ito two classes males ad females accordig to attribute sex. Similarly a group of idividuals may be classified ito smokers ad o-smokers with attribute smokig habit. If we classify group of items or uits or idividuals ito more tha tow classes the such classificatio is called maifold classificatio. e.g. group of persos may be classified accordig to their mother togue ito differet classes such as persos havig mother togue Marathi, Tamil, Telgu, Pujabi etc. I ay type of classificatio by attributes either dichotomy or maifold the importat thig is that the classes should be defied uambiguously. The classes should be mutually exclusive ad exhaustive. A item should belog to oe ad oly oe class ad o item should miss the classificatio..4. Classificatio of variables Whe the character uder study is quatitative type or variable the classificatio is doe accordig to values of variables. I case of discrete variables like chest size of baias i the stock held by hosiery shop, the variable assumes oly a few values like 6, 65, 7, 75, 8 -- cms. Here each possible value of variable forms a class. These classes are said to Mathematics & Statistics / 6

167 form discrete series of observatios o that variable. The o. of childre i family, o. of accidets i a day i a city, size of footwear etc. are some examples of variable which ca be classified i this way. There are geerally two types of variables we wat to study for geeral cosideratio. Suppose we have the followig iformatio about the umber of accidets that i a moth i a certai city We observed that the data we recorded is a discrete type meas there was umber of miimum accidet ad maximum 5 accidets. But there is o such case that accidet is.5 or -. This meas variable uder cosideratio is positive ad Iteger. So it is called discrete type ad the Distributio is called as, Ugrouped Frequecy Distributio Usig tally marks we write this iformatio i a tabular form as show i table- How to prepare frequecy distributio i) We fid miimum umber of accidets is zero ad maximum is 5. So first colum as Number of accidets ad values as,,,, 4, 5 ii) Next read the give observatio i the data ad make called a tally mark i the ext colum. Read all the data & make such marks. iii) Third colum as frequecy cout tally marks ad accordigly write umbers as frequecy. iv) Check whether we are correct or ot as the total of frequecy colum should be equal to total umber of observatios give i the data. Frequecy distributio for umber accidets Number of accidet Tally-marks Number of days Frequecy Grouped Frequecy Distributio (I) (II) Iclusive Method Exclusive Method Statistical Data / 6 Total This type of classificatio is most popular i practice. The weights i kg of 5 studets i a class are give i the followig data

168 To solve the above example. We observed the data Miimum umber as 46 ad Maximum as 64. Now we will classify this data accordig to class-itervals. We shall divide the umbers, i groups as 45-47, ) Class limits (class boudaries) There are two class limits lower ad upper class limit Class : Where 45 lower class limit 47 upper class limit. ) Class itervals (classes) 45-47, 48-5, are called class itervals or classes. ) Frequecy umber of observatios icluded i that class is called frequecy of that class. 4) Class width The differece betwee lower limit (lower boudary) ad upper limit (upper boudary) of that class is called class width. L Class width = upper limit lower limit e. g. class : 6-65 class width = upper limit lower limit = 65 6 = 5) Class marks (mid value) The arithmetic mea or average of the upper ad lower limits of a class is called class marks or mid value of that class (class iterval) lowerlimit upperlimit L the class mark (class mid value) = e.g. class class mark = = 46 (I) Iclusive Method The upper limit (boudary) is icluded i that class is called, Iclusive method of classificatio. e.g. class : 5-5 the values 5, 5, 5 are icluded. (II) Exclusive Method The upper limit is excluded i that class is called, exclusive method of classificatio e.g. class : the weight of studets upto 56 kg is added i this class but exact 56 kg are ot cosider it will cosider i the ext class as i Class boudaries I our example We cosidered weight of studets which is i kg. first class iterval is & secod is But if a studet has 47.5 kg weight. The we have to make classes as , thus all values betwee were cosidered, so this is called as class boudary. To get these class boudaries of a class, we add.5 to upper limit ad subtract.5 from the lower limit. Example : Fid class boudaries of the followig classes. Solutio : () 4, 4 9, 4,5 9, 4 () 4 6,6 8,8, 6,6 () Here iclusive method of classificatio is used. class boudaries are , , , , Mathematics & Statistics / 64

169 () Here exclusive method of classificatio is used Hece class boudaries are same as class limits. 4 6,6 8,8, 6,6 Frequecy Distributio of weights Class iterval Tally marks Frequecy Class boudaries Class marks Total 5 e.g. Followig is a frequecy distributio of o. of studets accordig to their pocket moey (i Rs.) Pocket moey No. of studets 7 5 I the above frequecy distributio variable uder study is pocket moey (i Rs.) of a studet ad method of classificatio is exclusive method. No. of studets belogig to respective classes are the class frequecies of those classes. Cumulative frequecies : Class frequecy is the o. of observatios i that class. But may times we may be iterested i kowig how may items have their values less tha (or more tha) the give value. e.g. We may be iterested i fidig o. of studets havig marks less tha 6 or o. of studets havig marks more tha 7. These umbers are also frequecies ad are called cumulative frequecies. The frequecies which give the o. of observatios less tha give value are called "less tha" cumulative frequecies ad those givig the o. of items havig values more tha the give value are called "more tha" cumulative frequecies. The computatios of less tha ad more tha frequecies is give below Height (i cms) No. of persos Less tha cumulative frequecies More tha cumulative frequecies The less tha cumulative frequecy of a give class is the o. of observatios havig their values less tha the upper boudary of the give class. I above example less tha cumulative frequecy of class 5 55 is 46 meas there are 46 persos havig height less tha 55 cms. Similarly more tha cumulative frequecy of a give class is the o. of observatios havig their values more tha the lower boudary of the give class Statistical Data / 65

170 I above example the more tha cumulative frequecy of class 5 55 is 5 meas there are 5 persos havig height more tha 5 cms. The table showig classes together with their cumulative frequecies is called cumulative frequecy distributio. Example: cms) Followig is a frequecy distribute of o. of screws accordig to the legth (i Fid less tha ad more tha cumulative frequecy distributio. Legth (i cms) No. of screws 7 9 Solutio : legth (i cms) No. of screws (frequecy) less tha cumulative more tha cumulative Check your Progress.4 Fill i the blaks. Colour of eyes is a. Variable.. Marks of studets is.. Variable. We select item radomly is samplig. 4. The umber of observatio belog to a particular class is called.. 5. Midpoit of a class iterval is called...5 GRAPHICAL REPRESENTATION OF DATA The frequecy distributio itself brigs out some importat features of raw data. However these features ca be studied more coveietly it we represet it i the diagramatic or graphical form. May questios about the data ca be aswered by meas of these graphs. The various types of graphs used for presetig frequecy distributio are (i) Histogram (ii) Frequecy polygo (iii) Ogive curves..5. Histogram This is simple method of represetig frequecy distributio graphically. I this graph classes are represeted by a series of adjacet rectagles. The base of each rectagle is the class iterval of that class. The area of each rectagle is proportioal to the frequecy of that class. Hece whe the class itervals are uiform throughout the distributio, the height of each rectagle is proportioal to the frequecy of correspodig class. Mathematics & Statistics / 66

171 But whe the class itervals are ot uiform the height of rectagle is proportioal to the frequecy desity of that class. Frequecy desity of class is ratio of class frequecy to its width. The histogram ca distiguish more clearly, class with maximum cocetratio of frequecy, This will be idetified by the rectagle with maximum height irrespective of the fact that the class itervals are equal or ot. It ca be used to determie mode of the distributio. I case of discrete frequecy distributio the rectagles are reduced to vertical lies as the class iterval are reduced to zero width. If class itervals are of type 5 9, 4, 5 9 etc. they are coverted ito cotiuous itervals by fidig class boudaries as , , respectively i order to have the rectagles adjacet to each other. Example : Followig is a frequecy distributio of o. of studets accordig to their marks i a test. Draw histogram for it. Marks No. of studets Histogram :.5. Frequecy Polygo Frequecy polygo is plotted represetig every class by a poit o a graph paper. The class mark or mid value of class iterval is take as X - co-ordiate ad the frequecy of the class as Y-co-ordiate of the poit represetig the class. Cosider two imagiary classes oe at each ed of the give distributio with frequecy zero. These are represeted by two poits o X axis oe at each ed. The cosecutive poits are the coected by segmets of straight lies. The figure eclosed by these lies ad the X-axis is i the form of polygo ad is called frequecy polygo. If the poits represetig differet classes are joied by a smooth curve the curve is called as frequecy curve. From both of these graphs we ca aswer the queries about symmetry of distributio the poits of maximum cocetratio of the frequecy ad the ature of frequecy distributio. Statistical Data / 67

172 Example : Represet followig frequecy distributio by meas of frequecy polygo. Class Frequecy Solutio : First fid the class marks of differet classes. Class Class Mark Frequecy Ogive curve Ogive curve is also called as cumulative frequecy curve. It is a smooth free had curve passig through the poits which have upper & lower class boudary as X-co-ordiate ad less tha (more tha) cumulative frequecy as Y-co-ordiate. Accordigly curve is called less tha or more tha Ogive curve. The less tha Ogive curve goes o risig from left to right ad o the other had move tha Ogive curve goes o decliig from left to right. The Ogive curves are very useful as we ca determie partitio values like media, quatities etc. from them. We ca also fid the umber ad percetage of observatios which lie betwee two give values of the variable. Example : Draw less tha Ogive curve for the followig distributio of daily wages (i Rs.) of workers i a small scale idustry. Mathematics & Statistics / 68

173 Wages No. of workers less tha cumulative frequecy Less tha Ogive curve No. of Workers Scale x axis : uit = 5 y axis : uit = Fig..4 Example : Draw more tha Ogive curve for the followig data. Class: Frequecy: 5 7 Solutio : Wages Class Frequecy less tha cumulative frequecy Statistical Data / 69

174 Check Your Progress -.5 Write True or False. Histogram is a simple method to represet a frequecy distributio.. Frequecy polygo is three dimesioal graph.. Ogive curve is also called as cumulative frequecy curve. 4. There are four types of Ogive curves 5. Frequecy polygo is a lie graph..6 DIAGRAMATIC REPRESENTATION OF DATA Frequecy distributio ca be represeted by a graph. But the categorical data caot be represeted by graphs. e.g. distributio of populatio of coutry accordig to religio caot be represeted by graph. Such data ca be represeted by meas of a diagram very attractively. The diagrams are easy to remember as they create loger lastig impressio o mid. Statistical data are made easily itelligece by meas of diagrams. Followig are some commoly used diagrams to represet statistical data. () Simple bar diagram () Sub divided bar diagram () Pie diagram.6. Simple Bar Diagram This is the simplest way of presetig the statistical data classified accordig to a sigle characteristic. It ca be used to preset data of populatio of differet cities, exports of differet coutries etc. It ca be used to represet ay sigle series but geerally it is used to show categorical series. I drawig simple bar diagram quatities are represeted by rectagular vertical bars separated from each other by uiform distace. The height of bar is proportioal to the magitude it represets. The width of bars must be the same for all bars as it does ot have ay sigificace. It is more coveiet to use graph for drawig a bar diagram. Usually values of variable are marked alog Y-axis ad the factor of classificatio or category are marked o X-axis. The scale of Y-axis must have zero as startig poit. This diagram is also kow as oe dimetioal diagram as it represets oly oe characteristic. Uless the order of bars has ay sigificace it is suggested that the bars should be arraged i icreasig or decreasig order of magitudes represeted by them. This makes the diagram more attractive as well as it facilitates the compariso. Mathematics & Statistics / 7

175 Example : Solutio : Preset the followig iformatio by bar diagram Coutry Birth rate (per ) Ira Libya Malaysia 4 Mexico Swede 5.6. Subdivided bar diagram I may cases we have to represet a whole quatity ad its sub divisios i the same diagram. I that case we ca use bar diagram to represet whole quatities ad the sub divisio ca be represeted proportioally by dividig each bar ito umber of parts. This type of bar diagram is called subdivided bar diagram. The subdivided bar diagram is draw usig followig steps. () Draw oe bar for each of the whole quatity with its height proportioal to the magitude it represets. () Usig same scale divide each bar ito differet parts proportioally. The order of subdivisios from bottom to top should be the same i each bar. () Use differet otatios like horizotal, vertical or slatig lies or dots or colums for showig subdivisios. (4) Give the title, scale ad explaatio of otatio used at a suitable place i the diagram. Subdivided bar diagram ca be used () to represet data of populatio of differet states i coutry with its subdivisios accordig to religio () to represet data of studets studyig i college for o. of years with subdivisios accordig to classes. Example: Represet followig data by sub divided bar diagram Year No. of Studets FY SY TY Total Statistical Data / 7

176 .6. Pie diagram Pie diagram is special type of diagram used to represet whole quatity by a circle ad subdivisio of whole quatity are show by sectors of that circle. The whole circle is divided ito differet sectors, areas of which are proportioal to the magitudes they represet. It is very easy to divide circle ito sectors as the area of each sector is proportioal to agle it subteds at the cetre. Hece to divide the circle ito sectors reduces to divide agle of 6 ito proportioal parts. The agle for a particular sector is give by the relatio = partial quatity Total quatity 6 This diagram is a two dimesioal diagram because i this case area of sector represets the quatity. Pie diagram ca be used to represet the subdivisios of total budget or total expediture or total icome etc. The ame pie diagram is derived from the word pie which meas a cake or slice of it with layer of custard o it. For drawig a pie diagram the first step is to covert all the sub quatities ito agle usig above formula. The draw a circle of suitable size. Start measurig agles from some referece lie from cetre to circumstaces. These agles divide the circle ito differet sectors. We may mark the sectors by differet sigs such as dots, crosses, parallel lie or differet colors, quatities represeted by sectors may be writte iside the sectors of the circle. Example : Draw Pie diagram for the followig data o percetage of expediture o differet items i a average family budget. Items : Food Ret Clothig Fuel Others % expediture: Solutio : The first step is to covert the quatities ito proportioal agles. e.g. agle for the item food is 4 = 6 = 44 The agles for remaiig items are obtaied as follows. Item % expediture agle food 4 44 Ret 7 Clothig 5 54 Fuel 6 Others 5 54 The Pie diagram represetig these data is give below Mathematics & Statistics / 7

177 Illustrative examples. Example Frequecy distributio of scores of 8 cadidates is give below. Score No. of cadidates () Fid class boudaries of all the classes. () What is the lower class boudary of 4th class? () What is width of rd class? (4) What is the class mark of 6th class? (5) Fid less tha cumulative frequecies. Solutio : Class Class Frequecy Frequecy Statistical Data / 7 Less tha cumulative Frequecy () Class boudaries are obtaied i above table () Lower boudary of 4th class = 89.5 () Width of rd class = = (4) Class mark of 6th class = 9 = 4.5

178 (5) Less tha cumulative frequecies are obtaied i above table Example Draw histogram ad frequecy polygo for the followig data. Size of farm (i hectors) No. of farms Solutio : Here first step is to fid class boudaries Size of farms No. of farms Class boudaries Class mark Example Draw less tha ad more tha Ogive curves for the followig data class Frequecy Class Frequecy Solutio : First we shall fid less tha ad more tha cumulative frequecies Class Frequecy Less tha C.F. Less tha C.F Mathematics & Statistics / 74

179 Example 4 : Draw simple bar diagram for the followig data o o. of studets erolls or certai course for differet years Year No. of studets Solutio: Examples 5 : Preset the followig data usig suitable diagram Class F.Y. S.Y T.Y Pass 5 Fail 5 8 Total Statistical Data / 75

180 Solutio : Example 6 : Draw Pie diagram for the followig data Coutry Idia Sri Laka USA U.K. Mexico Populatio growth rate (%) Solutio : Here first step is to fid the agles for differet coutries Coutry Populatio agle growth rate (%) Idia.. 6=7. Sri Laka.8.8 6=58.9. USA.. 6= UK.8.8 6=58.9. Mexico.. 6=4.7. Total SUMMARY Graphs is very strog statistical tools for presetig a give frequecy data. We collect a data with differet methods. From the data we draw sample with radom samplig the process the raw data. Calculate measures of cetral tedecy a the with the help of give type of decided variable we draw particular types of graphs to get values of variables. Mathematics & Statistics / 76

181 .4. Qualitative. Quatitative.8 CHECK YOUR PROGRESS ANSWERS. Simple radom samplig 4. Class frequecy 5. Class marks..5. True. False. True 4. False 5. False.9 QUESTIONS FOR SELF - STUDY. Explai differet methods of classificatio briefly. Give suitable examples.. Explai the followig terms with illustratios. (i) Attribute (ii) Variable (iii) Class limits (iv) Class width (v) Class frequecy (vi) Class mark (vii) less tha ad more tha cumulative frequecy. Followig is a frequecy distributio of heights i cm. Height No. of persos (i) Fid class boudaries of each class. (ii) Determie class width of each class (iii) Fid less tha ad more tha cumulative frequecies 4. Write a short otes o (i) Histogram (ii) frequecy polygo (iii) less tha Ogive curve (iv) more tha Ogive curve 5. Draw histogram ad frequecy polygo for the followig data Class Frequecy Statistical Data / 77

182 6. Draw less tha ad more tha Ogive curves for the followig frequecy distributio. Marks No. of studets Draw histogram for the followig data Weight (i Kg) No. of studets Frequecy distributio of screws accordig to their legth i cms is give below legth i cm. No. of screws (i) Determie class boudaries of all the classes (ii) What is the width of 4th class? (iii) Fid less tha cumulative frequecies (iv) Draw more tha Ogive curve. (v) Draw histogram 9. Draw suitable diagrams i each of the followig cases (i) Followig are the result of survey regardig viewership of differet histograms telecast by Doordarsha Mahabharata 96 % Hidi film 65 % Chitrahar 55 % Ragoli 6 % Hidi Serials 5 % Hidi News 5 % Eglish News % Mathematics & Statistics / 78

183 (ii) The followig table shows the cost of goods produces i a factory for differet year Year Cost of goods (i Rs.) (iii) Category Reveue i % to total Corporate tax 4.5 Icome tax 5. Excise duty 9.5 custom. (iv) Year No. of studet i course MCM MCA BCA Total MCM (v) Item Food clothig Recreatio house ret expediture Write short otes o (i) Simple bar diagram (ii) Subdivided bar diagram (iii) Pie- diagram. Costruct subdivided bar diagram for the followig data Year Import Export Statistical Data / 79

184 . By the Ecoomic budget of Maharashtra state of -4, Oe Rupee comes from ad oe Rupee goes to is give below- Rupee comes from No. Tax & Reveue Amout (Rs.) Iteral debt of the state 4.4% State s ow tax reveue 55.% Loas ad advace by state govermet.4% 4 Grats-i-aid from cetral govermet 9.59% 5 Share of cetral taxes 9.% 6 State s ow otax reveue 6.7% 7 Public accout.4% 8 Loas from cetral govermet.4% Rupee goes to No. Tax ad Grats Amout (Rs.) Social Service 7.8% Grats-i-aid to Local bodies.8% Loas ad advaces give by state govermet.64 4 Repaymet of public debt 6.77% 5 Iterest paymet ad debt services.69% 6 Capital expediture.% 7 Ecoomic Services.69% 8 Geeral Services 7.%. SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi Mathematics & Statistics / 8

185 NOTES Statistical Data / 8

186 NOTES Mathematics & Statistics / 8

187 Chapter Measures of Cetral Tedecy. Objectives. Itroductio. Arithmetic mea.. Properties of arithmetic mea.. Merits ad Demerits of mea. Media.. Merits ad Demerits of media.4 Mode.4. Merits ad Demerits of mode.5 Summary.6 Check your Progress - Aswers.7 Illustrative Examples.8 Questios for Self Study.9 Suggested Readigs. OBJECTIVES After studyig this chapter you will be able to Explai what is Mea Discuss how mode is calculated Calculate Media Discuss about cetral Tedecy Discuss about value of Cetral item. Explai values comig agai ad agai. INTRODUCTION We have studied i the previous chapters that the first step towards codesatio of raw ad large data ito compact form is to classify it ad prepare frequecy distributio. I the form of frequecy distributio of data it becomes easy to uderstad may features of data such as patter of variatio of values, portio of cocetratio of values, symmetry of distributio etc. It is a descriptive measure as it depicts the patter of behaviour of the variable. However for further statistical aalysis we eed the data to be codesed or summarized ito a sigle umber which may be take as the represetative umber of the whole group. Such a umber is called as a average or cetral value. I most of the data we ote a property of observatios or values to cocetrate i a cetral part of data. I other words large proportio of observatios are gathered ear cetral value. This property of observatios i a data is called as property of cetral tedecy. Naturally we select a represetative observatio from the cetral part ad such observatio i cetral part aroud which large o. of observatios i a data are cocetrated is called measure of cetral tedecy. I most of data average is a cetre of cocetratio of values. I that sese average is called measure of cetral tedecy. The average locates cetre of data ad i may cases the whole distributio is idetified by the average. The average is therefore called measure of locatio. Average is a descriptive measure ad it ca focus attetio more sharply o various properties of data. There are may types of averages each havig particular properties ad each beig typical or represeted i some uique way. The most frequetly used averages are the arithmetic mea, the media ad the mode. Measures of Cetral Tedecy / 8

188 . ARITHMETIC MEAN OR MEAN This is most commoly used ad widely applicable average. Mea is a familiar average to a comma ma. Defiitio : Mea is defied as the ratio of sum of observatios i the data to the umber of observatios. Computatioal Formula I statistics while computig differet measures for the data o variable we require to cosider two types of data. (i) Ugrouped data (discrete variables) (ii) Grouped data or frequecy distributio (cotiuous variable) Accordigly the computatioal formula for these two types of data are differet. (I) Ugrouped data Suppose that x x , x are the give observatios. The mea of these observatios is deoted by x (read as x bar) ad is give by X sum of observatios = o. of observatios x x... x X = X x i = Example : Aual sales (i, Rs) of a compay for moths are give below., 47, 9,, 5,, 4,, 5, 5. Fid mea aual sales. Solutio : Here there are = observatios x x...x X = =. Mea aual sales = Rs. = (II) Grouped data (frequecy distributio) I case of frequecy distributio suppose k = o. of classes X i = class mark of i th class f i = frequecy of i th class i =,...k The mea of frequecy distributio is deoted by x ad is give by x X f x f = f f X = f x i f i i... x... f k k f k Mathematics & Statistics / 84

189 Steps for fidig mea () Fid class marks x i - values of all the classes. i =,...k () Fid values of X i f i i =,...k () Usig formula fid x Example : give below Frequecy distributio of marks obtaied by studets is Fid mea marks Marks No. of studets Solutio : First it is required to fid class marks of all classes Marks No. of studets (f) class mark xi fi-xi =5 5 = = = Total 44 Mea x = f i f x Mea marks is 4.4 i i 44 = =4.4.. Properties of arithmetic mea ) If we kow the umber of values i the data ad mea x the we ca fid sum of values i the data. sum of values = x x ) The sum of deviatios of values i the data from its mea is equal to zero. If x, x,..., x are observatios ad x is their mea. The x x, x x, x x are deviatios of these observatios from their mea. x i x x i x x x ) Effect of chage of origi ad scale let x, x,. x be give observatios ad x is their mea. If we trasform x i to u i i =, usig chage of origi ad scale as Measures of Cetral Tedecy / 85

190 u i = x i a 4 i =,... a, are costats the the mea u of u, u,,u is give by x a u so that x a u This result is useful for simplifyig the computatios of mea particularly whe observatios are large ad i case of frequecy distributio. year. Example : Followig are data o o. of studets i differet colleges i first 5,, 98,, 5,,, 6 calculate mea o of studets Solutio : Here observatios are large so we trasform them to ew observatio by subtractig from each observatio a suitable costat The ew observatios are u i x i a i =, 8 =, a =, u = 5, u =, u =, u 4 =, u 5 = 5 u 6 = u 7 =, u 8 = 6 u i 4 u 5. 8 Example 4 : mea x = a + u = + 5.=5. For the followig frequecy distributio o heights of studets compute mea height. Height (i cms) No. of studets usig Solutio : We shall solve this problem by trasformig class marks (x i ) to u i Height No. of studets (fi) Class mark x i 45 u i f (X i ) i u i Total 6 Mathematics & Statistics / 86

191 chage of origi ad scale u = f i u f i = 6 =.4 6 x=a+u = 45.4.= Merits ad Demerits of mea The cocept of mea is familiar ad log usage ad hece it seems to be best average or best measure of cetral tedecy. Moreover there are certai limitatios i usig it. Followig are merits ad demerits of mea as a measure of cetral tedecy. Merits : () It is rigidly defied ad uiquely determied. () It is familiar to commo ma ad easy to compute. ()!t is based o all values i the data ad therefore is more stable. (4) It is capable of further algebraic treatmet. (5) It is least affected by samplig fluctuatios. (6) It is widely used i practice ad is most commoly used average i may fields. (7) Observatios eed ot required to be arraged i order for computatios of mea. Demerits : () The mea ca be used oly whe characteristic uder study is a variable. For attribute type character mea ca ot be determie. () It is much affected by extreme observatios specially whe o. of observatios i the data is small. () If frequecy distributio is havig ope ed classes mea caot be determied. Because we caot fid class mark value for such ope ed classes. (4) It caot be determied graphically.. THE MEDIAN The media of data is the value of cetral item or observatios whe the observatios are arraged i ascedig order of magitude. For most of the data the media ca serve as a average as it will be always located at cetre of the data. It is a positioal average. There are equal o. of observatios above ad below the media i the data. It divides the data ito two equal parts. It is the most suitable measure of cetral tedecy for distributio's like icome distributio or age distributio which are mostly o-symmetric. Computatioal formula (I) Ugrouped data : I case of ugrouped data whe observatios are give media = value of icreasig order ( is odd) th observatio whe observatios are arraged i th media = mea of ad observatio whe observatios are arraged i ascedig order. ( is eve) Example 5 :- Compute media of followig observatios i each set (i),,,7,,9,7 (ii)7,,4, 5, 7, 8, 8, 6 th Measures of Cetral Tedecy / 87

192 Solutio () Here o. of observatios = 7 which is odd o. First we arrage observatios i ascedig order. 9,,,,7,7, media = value of ( 7 ) th = 4 th observatio = (ii) Here o. of observatios = 8 which is eve o. first we arrage observatios i ascedig order. 8,6,7,,5,8,4 Media = mea of ( ) th = 4 th ad ( +) th = 5 th observatios = 5 (II) Grouped data : (Frequecy distributio) = I case of grouped data media is give by the followig formula. Media = L + [ N cf] f h L = lower class boudary of media class N = Total frequecy = f i c. f = less tha cumulative frequecy of the media class. h = width of media class. f = frequecy of media class. Media class : It is the class havig less tha cumulative frequecy just greater tha or equal to N/ Steps () first fid less tha cumulative frequecies of all the classes. () Determie media class. () Determie the values of L, c. f., h, f, (4) Usig the formula compute media Ratable No. of Value (Rs) dwellig Example 6 : Frequecy distributio of ratable value of dwellig i locality is give below. Mathematics & Statistics / 88

193 Solutios : Ratable No. of value (Rs.) less tha cumulative dwelligs frequecy Total N = = for the class less tha cumulative frequecy is just greater tha N = (Note that for classes owards less tha cumulative frequecy is greater tha ) Media class is L =, c.f. = 85, h =, f = 85 Media = L + [ N cf] f h = + [ 85] 85 = = = Rs = 94. Rs... Merits ad Demerits of media : Wheever the mea fails to be a good measure of cetral tedecy the media i geeral is foud to be useful ad the appropriate average. It has several advatages ad limitatios also. Followig are merits ad demerits of media Merits () It is applicable to all kids of data o variable or attributes. I case of qualitative data the items ca be arraged i particular order accordig to a qualitative character ad the quality of cetral item gives the media or average quality. () For o-symmetric distributios like age distributio or icome distributio the media is most appropriate average. () It is ot affected much by extreme observatios i the data. (4) Cocept of media is easy to uderstad ad is appealig. (5) It ca be determied eve if there are ope ed classes i case of frequecy distributio (6) It is least affected by choice of class itervals. (7) It is useful whe the mea is either idetermiate or usuitable. (8) It ca be determied graphically. Demerits : () It is ot based o all the observatio i the data. () It is ot as rigidly defied as the mea. Measures of Cetral Tedecy / 89

194 () It is ot suitable average from small group of item. (4) It is less capable of further mathematical treatmet. (5) It eeds to arrage observatios i ascedig order..4 THE MODE The word mode meas fashio. We say that wearig arrow bottom trousers is the curret fashio amog yougsters. It meas that majority of yougsters wear that type of trousers. The mode Mo is thus defied as the value of the variable occurrig more or maximum o. of times i the data tha ay other value. It is the most frequetly occurrig value i the data. Computatio of mode: (I) Ugrouped data: I case of ugrouped data of observatios x, x, x mode is that observatio which occurs maximum umber of times. Example7 : Calculate mode of the followig observatios.,,7,,7,5,7,,7,9. Solutio : I the above observatios observatio 7 occurs more o. of times as compared to other observatios. Hece mode is 7. (II) Grouped data (Frequecy distributio) : I case of frequecy distributio mode is give by the followig formula. f f Mode = L + xh f f f L = lower class boudary of modal class. f = frequecy of pre-modal class f = frequecy of modal class f = frequecy of post modal class h = width of modal class. Modal class = It is the class havig maximum frequecy Steps () Determie modal class () Determie the values of f, f ; f, h, L () Usig the formula determie the value of mode. Example 8 : The marks obtaied by 4 studets i a certai test is give below. Fid model marks. Solutio No. of studets Marks Marks No. of studets Mathematics & Statistics / 9

195 Here maximum frequecy 6 correspodig to class modal class is L =, f =, f =6, f = 8h = Mode = L + f = + = + =.85 f f f f xh = Merits ad Demerits of mode As compared to mea ad media the mode has very limited utility. Followig are merits ad demerits of mode. Demerits : () It is ot based o all observatios i the data ad hece it is ot sesitive to the chages i extreme values i the data. () It is ot suitable average whe the umber of items i the data is very small. () It is ot suitable average for extremely o-symmetric distributios. (4) It caot be determied whe maximum frequecy is at oe ed of distributio. (5) It is affected to a great extet by the choice of class itervals. Merits : () It is applicable to both qualitative ad quatitative type data. () It is useful i some special type of situatios oly. () It is ot iflueced by extreme values i the data. (4) It ca be determied graphically Check Your Progress -. to.4. What is mea?. What is media?. What is mode? 4. Choose the correct alterative. i. For a set of distict values, The media value happeed to be 55. Later it was observed that a value 74 was wrogly writte as 64. With this correctio ow a) The media will udergo a chage ad gets icreased. b) The media will udergo a chage ad gets decreased. c) The media will be uchaged. d) The give iformatio is isufficiet for recalculatio of media. Measures of Cetral Tedecy / 9

196 ii. For the followig distributio, how would the mea compare with the media? iii. iv. a) The Mea would be less tha the Media b) The Mea would be equal to the Media c) The Mea would be greater tha the Media d) Noe of the above If a costat value 5 is subtracted from each observatio of a set, the mea of the set is a) icreased by 5 b) decreased by 5 c) ot affected d) 5 times the origial value A distributio of 6 scores has a media of. If the highest score icreases by poits, the media will become a) b).5 c) 4 d) caot be determied without additioal iformatio v. The value of (x i -x)/, is a) zero if x = x i i b) always zero c) d) oe of the above.5 SUMMARY Mea is othig but average which is the ratio of sum of observatios i the data to the umber of observatios. It is deoted as x (read as x bar) Mea is calculated o two types of data. Ugrouped Grouped for ugrouped data is formula is x Where x i = x + x +..x x i = No. of observatio for Grouped data the formula is x Where f i x i = x f +x f + +x k f k i f x f i = f +f + f k Where k = o. of classes f i i X i = class mark of i th class i =,..k f i = frequecy of i th class Mathematics & Statistics / 9

197 The Media of data is the value of cetral item of observatios whe the observatios are arraged i ascedig order of magitede. Formula of Medio for (I) Ugrouped data : I case of ugrouped data whe observatios are give media = value of ( ) icreasig order ( is odd) th observatio whe observatios are arraged i th th media = mea of ad ( ) observatio whe observatios are arraged i ascedig order. ( is eve) (II) Grouped data : (Frequecy distributio) I case of grouped data media is give by the followig formula. N h Media = L + [ cf ] f L = lower class boudary of media class N = Total frequecy =f i c. f = less tha cumulative frequecy of the media class. h = width of media class. f = frequecy of media class. Media class : It is the class havig less tha cumulative frequecy just greater tha or equal to N/ Steps () first fid less tha cumulative frequecies of all the classes. () Determie media class. () Determie the values of L, c. f., h, f, (4) Usig the formula compute media - The Mode is the value of the variable occurrig more or maximum o. of times i the data tha ay other value. Grouped data (Frequecy distributio): I case of frequecy distributio mode is give by the followig formula. Steps () f f Mode = L + f f f h L = lower class boudary of modal class. f = frequecy of pre-modal class f = frequecy of modal class f = frequecy of post modal class h = width of modal class. Modal class = It is the class havig maximum frequecy Determie modal class () Determie the values of f, f, f, h, L () Usig the formula determie the value of mode. Ugrouped data: I case of ugrouped data of observatio x x x mode is that observatio which occurs maximum umber of times. Measures of Cetral Tedecy / 9

198 . to.5.6 CHECK YOUR PROGRESS - ANSWERS. Mea is othig but average which is the ratio of sum of observatios i the data to the umber of observatios. It is deoted as x (read as x bar). The Media of data is the value of cetral item of observatios whe the observatios are arraged i ascedig order of magitude.. The Mode is the value of the variable occurrig more or maximum o. of times i the data tha ay other value. 4. (i) c, (ii) b, (iii) b, (iv) a (v) - a.7 ILLUSTRATIVE EXAMPLES Example : The startig mothly salaries of employees recruited i a firm are Rs. 5, 75, 68, 8, 85, 75,, 75, 575 ad 75 Fid the mea, media ad the mode Solutio : Let x, x.. x be give observatios Mea : Mea x = x i 74 =74Rs Media : For fidig media we arrage observatios i ascedig order. 5, 575, 68, 75, 75, 75, 75, 8, 85, Here o. of observatios = Media = mea of 5 th ad 6 th observatio = 75 Rs. Mode : Observatio 75 is repeated maximum o. of times. mode = 75 Example The distributio of life time i hrs. of radio tubes is give below. Calculate the mea, media ad the mode Life Tubes Solutio : fu 7 Mea : u =. 5 Mathematics & Statistics / 94

199 Life (i hrs) No. of tubes (f i ) Class mark x i less tha x u i i cf f i u i Media Total N = 7 f x = a + hu a = h = x = x.5 = hrs. From less tha cumulative frequecy observe that for the class the less tha cumulative frequecy is just greater tha N = Media class is L = c. f =95 h = f = 6 Media =L+[ N cf] f h = + [ 95] 6 = hrs. Mode : Maximum frequecy correspods to class Modal class is L = f =55 f = 6 f = 5 h = f f Media = L + [ f f f ] h 6 55 = + [ ] 55 5 = 5 hrs.8 QUESTIONS FOR SELF - STUDY. Defie mea ad state its importat properties.. Defie media. State merits ad demerits of media.. Defie mode. State merits ad demerits of mode. 4. What do you mea by cetral tedecy of data? What is measure of cetral tedecy? 5. State merits ad demerits of mea. 6. The lifetime i days of 8 small isects is give below. 5, 4,8, 9,6, 7, 5, 7. Fid mea ad media life time. 8. A icomplete frequecy distributio is give below. The total frequecy is ad the media is 46. Fid the missig frequecies. Measures of Cetral Tedecy / 95

200 Marks : Studets? 65? 4 9. The frequecy distributio of o. of tablets required to cure fever is give below. Fid the mea, the media ad the mode Tablets No. of persos The followig is the age distributio of life isurace policy holders whose mea age is.6 years. Fid the missig frequecies. Age: Persos: 7?. The mothly expediture (i Rs.) of families is give below. Fid mea, media ad mode. 7, 75, 7, 8, 75, 775, 8, 75, 7, 75.9 SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi Mathematics & Statistics / 96

201 NOTES Measures of Dispersio / 97

202 NOTES Mathematics & Statistics / 98

203 CHAPTER 4 Measures of Dispersio 4. Objectives 4. Itroductio 4. Rage 4. Mea Deviatio 4.4 Variace 4.5 Stadard Deviatio 4.6 Absolute ad Relative Measure of Dispersio 4.7 Coefficiet of Variatio 4.8 Summary 4.9 Check Your Progress - Aswers 4. Illustrative Example 4. Questios for Self - Study 4. Suggested Readigs 4. OBJECTIVES Frieds, Dispersio meas the expase of the give sample data. After studyig this chapter you will be able to Explai Rage Discuss Mea Calculate Variace 4. INTRODUCTION The average or measure of Cetral tedecy is a good descriptive measure of a distributio of a variable. However it caot describe the distributio completely. It gives us idea about the locatio of Cetre of the distributio. For complete kowledge of the distributio some additioal iformatio is required. Oe such iformatio is that about ature ad extet of variatio of the values i the data. The average scoress of two batsme for a seaso may be equal or early equal but their cosistecy ca be judged by studyig the variability of their scores e.g. If the scores of oe batsma are 4, 45, 5, 56, 59 ad those of other are, 5, 5, 6, 85 the they do ot differ i average but they differ i variatio. Hece oly average is ot sufficiet for comparig the performace of two batsme. This ature ad extet of variatio of values i the data is kow as dispersio. The kowledge of dispersio helps i judgig the reliability of the average. The average of the data will be more reliable or represetative of data whe the data has less variability. This aalysis of variatio i values i the data has o of practical applicatios i various fields like agriculture, idustry medical etc. For the study of dispersio preset i the data we eed some measure of the degree of dispersio ad it is called measure of dispersio. I the remaiig chapter we are goig to study some measure of dispersio. 4. RANGE Rage is simplest measure amog several measures of dispersio. Rage is defied as the differece betwee maximum ad miimum observatios i the data. I case of frequecy distributio rage may be defied as the differece betwee smallest ad largest class boudaries. Sice rage is the simplest measure to Compute, it is the crude measure of dispersio. The rage is used i limited applicatios ad also it has certai defects. It is greatly affected by a uusal value of the extremity. We ca ot iterpret the value of rage properly without kowig the o. of observatio. Measures of Dispersio / 99

204 The rage is useful is situatios where oe desires to kow oly the extet of extreme dispersio. The stock market reports are frequetly stated i terms of their rage by quotig the high ad low price of stock over a period. I weather reports also we fid maximum ad miimum temperatures stated. The daily mea temperature ca be obtaied by averagig these two temperatures. I quality cotral rage is used as a measure of variatio withi the sample. The rage beig easy to compute ad is a commo way of describig dispersio is ofte used i egieerig ad medical reports. Illustratios () : Followig are the prices of stock market shares of a certai compay for last days. Fid the rage, 98, 96,, 5,, 7, 5,, 99 Aswer : Here miimum observatio is 96 ad maximum observatio is. Hece rage is 96 = MEAN DEVIATION Rage as a measure of dispersio does ot takes ito cosideratio all observatios i the data. So it is Comparatively ustable ad isesitive measure of dispersio. Hece it is ot further useful for aalysis of data. Mea deviatio is a measure of dispersio based o all observatios i the data. By deviatio we mea subtractig same costat from give observatio ad is called deviatio of that observatio from that costat e.g. deviatio of x from arbitrary costat a is x a. I mea deviatio we cosider the deviatio of each observatio from some costat a. The mea of absolute deviatios of observatios from a is called mea deviatio about 'a'. Defiitio. (I) I case of observatio x, x, x the mea deviatio about a is g x i a MD about a= (II) I case of frequecy distributio x i = class mark of i th class i =,, k f i = frequecy of i th class mea deviatio about a is give by MD about a = f x i i f a i Usually we obtai mea deviatio about some cetral value such as mea or media or mode accordigly we get mea deviatio about mea or mea deviatio about media or mea deviatio about mode. Illustratio : () Calculate mea deviatio about mea for the followig observatio. 5, 7,, 8, 6,, 4,, 5, Aswer : x i x i x Total 7 6 MD about mea = xi x mea 7 7 x i x 6.6 Mathematics & Statistics /

205 () Frequecy distributio of umber of days of medical leaves ejoyed by employees is give below. No. of Days No. of Employees Calculates mea deviatio about mea. Solutio : No. of Days No. of Employees f i x i f i x i x i x f i x x i Total 7 98 x 7 = 4. MD about mea = f i x i f x i 98 = = 9.9 days 4.4 VARIANCE Variace is the mea square deviatio about mea. Thus variace is defied as the mea of square of deviatios take from arithmetic mea. Variace is good measure of dispersio ad it has may desirable properties. It is deoted by (sigma squared) (I) I case of observatios x x x the variace is defied as = x Computatioal formula x i x x where x = mea = x x i i i i = x = x i x x x x (II) = x = i x x x I case of frequecy distributio. X = Class mark of i th class I =,, k Measures of Dispersio /

206 f i = frequecy of i th class Variace is defied as f I x i x = Computatioal Formula f i where x = mea = f i f f i x i x f i x i = i x.x x f ix f = i i f f i f ix x f fi xi i x i i x f i f i x f I case of frequecy distributio as well as idividual observatios calculatios of variace ca be simplifies by makig use of charge of origi ad scale. Chage of origi ad scale : i i x x Let u i = i A be the ew observatios obtaied from xi h by usig charge of origi ad scale so that - X i = A +hu i x = A + h u the variace of u is give by u = u i u (i case of idividual observatio) f u f i i i ui f iu i whe u ad u = u whe u zf f i f i (i case of frequecy distributio) The variace of origial observatio is x = h u Illustratios (I) Followigs are mothly savigs (i Rs.) of families. 5, 75, 7,, 8, 9, 7, 84, 98. Solutio : Fid variace. x x Total x x 849 = = 849 = x i i x = =98 Mathematics & Statistics /

207 () Fid the variace of the followig distributio of percetage divided paid by 5 compay. Solutio : Divide d No. of Compaies (f) Divided No. of Compaies Class mark X u = x 5 6 Measures of Dispersio / Fu fu Total x i 5 u i 6 x i = 5+6u i = A+hu gives A = 5 h = 6 fu 4 u. 8 f 5 x = A + h u = 5 + 6(.8) = 4.5 =.8 x = h u u = fu 74 u (.8) f 5 x = STANDARD DEVIATION Stadard deviatio is most Commoly used measure of dispersio. It has bee devised to remove the drawback of the variace that it is rather a awkward value to iterpret. The uits attached to variace are squares of the uits i practice. e.g. cm, Rs etc. But we defie stadard deviatio as the positive square root of variace or the root mea square deviatio from the arithmetic mea. Due to this stadard deviatio ca be expressed i the same uits as that of the origial data. It also has all advatages of the variace as a measure of dispersio. However from magitude of stadard deviatio we caot immediately say whether distributio has small or high degree of variability. Stadard deviatio is deoted by or SD. Formula of Computig SD are as follows : = x x where x x (i case of observatios x, x, x ) = f i f x i x f i x where x f (i case of frequecy distributio) i i

208 4.6 ABSOLUTE AND RELATIVE MEASURES OF DISPERSION The measure of dispersio like rage, mea deviatio, variace, stadard deviatio measures the magitude of dispersio ad they are called measure of absolute dispersio. These are to be expressed with appropriate uits. They are useful for compariso of variability of two sets of data oly whe both are i the same uits ad their cetral values ie. averages are early equal. But i may problems situatios oe or both of these coditios are ot fulfilled. So we eed measures of dispersio which are idepedet of uits. Such a measure ca be obtaied by takig ratio of the absolute measure of dispersio to same cetral value of the data. It is called measure of relative dispersio. Most commoly used measure of relative dispersio is coefficiet of variatio. 4.7 COEFFICIENT OF VARIATION Coefficiet of variatio (cv) is widely ad commoly used measure of dispersio. Wheever we require to compare the variability of sets of values we use cv. It is defied as the ratio of stadard devidatio of the series to its arithmetic mea. It is always expressed i percetage. SD CV = mea The series which has less CV is said to be more cosistet or stable. Check Your Progress - 4. to 4.6. Defie the followig terms. (a) Rage (b) Mea deviatio (c) Variace (d) Stadard deviatio. Choose the correct aswer from give. i) Let A = {,,,,} ad B = { 4,,,, 4} Let m(.) ad v(.) deote the mea ad variace respectively of the set metioed. The idetify the correct statemet. a) m(a) > m(b), v(a) < v(b) b) m(a) > m(b), v(a) < v(b) c) m(a) = m(b), v(b) = 4v(A) d) m(a) = m(b), v(b) = v(a) Mathematics & Statistics / 4

209 ii) Which of the followig measures of dispersio does ot deped o the uits of measuremet? a) S. D. b) Mea Deviatio c) Rage d) C. V. iii) Mea deviatio is miimum whe the observatios are take from a) Mea b) Media c) Mode d) Q4 iv) If you are told that a populatio has a mea of 5 ad variace of zero what must you coclude? a) Someoe has made a mistake b) There is oly oe elemet i the populatio c) There are o elemets i the populatio d) All the elemets i the populatio are 5 v) The followig set of scores is obtaied o a test X : 4, 6, 8, 9,,, 6, 4, 4, 4, 6. The teacher computes all of the descriptive idices of cetral tedecy ad variability o these data, the he discovered that a error was made ad oe of the 4's is actually a 7. Which of the followig will be chaged from the origial computatio? a) media b) rage c) S. D. d) Noe of the above vi) If each observatio of a series is multiplied by a costat C, the coefficiet of variatio as compared to the origial value a) is icreased by C b) is decreased by C c) remais uchaged d) is C times the origial value 4.8 SUMMARY Variatios of values i the data is kow as dispersio. Measure of dispersios are Rage, Mea deviatio, Variace, Stadard deviatio. Rage is defied as differece betwee maximum ad miimum observatios. Mea deviatio is a measure of dispersio based o all observatios i the data which is calculated by subtractig same costat from give observatio ad is called as deviatio of that observatio from that costat. Variace is the mea square deviatio about mea. Variace is defied as the mea of square of deviatios take from arithmetic mea. Stadard deviatio is the positive square root of variace or the root mea square deviatio from the arithmetic mea. Rage, mea deviatio, variace ad stadard deviatio are called as absolute dispersio. 4. to CHECK YOUR PROGRESS- ANSWERS. (a) Rage is defied as differece betwee maximum ad miimum observatios. (b) Mea deviatio is a measure of dispersio based o all observatios i the data which is calculated by subtractig same costat from give observatio ad is called as deviatio of that observatio from that costat. Measures of Dispersio / 5

210 (c) (d) Variace is the mea square deviatio about mea. Variace is defied as the mea of square of deviatios take from arithmetic mea. stadard deviatio is the positive square root of variace or the root mea square deviatio from the arithmetic mea. (i) c, (ii) d, (iii) b, (iv) d (v) c, (vi) d 4. ILLUSTRATIVE EXAMPLES Example I: The scores of batsme i a certai test are as give below : 5, 47, 5, 45, 6, 7, 4, 58 Fid (i) variace (ii) coefficiet of variatio Solutio : Here data give are 8 observatios Say x,x,.. x 8 We use chage of origi for simplifyig calculatio as u i = x 45 i =,... 8 So that ew observatios ad their squares are as follows : X i u i = X i 45 u i Total Example : The umber of items of a idustrial product sold by two salesma A ad B i te moths i a year are give below. From these date determie which salesma is more cosistet. Number of Items Sold u i 5 u = x = mea = 45 + u = u = variace for u u = i u 667 (.875) 8 = x = u = , x = + = x A B Solutio : For judgig which salesma is more cosistet we have to compare the variability of their performace. This ca be doe more appropriately by compariso of their coefficiets of variatio. So let us fid the mea ad the SD for each of the two series, here the give values are large i size. So we may use the method of chage of origi. Number of items sold Salesma A Salesma B x y u= x - 4 u v = y 4 V Mathematics & Statistics / 6

211 : The computatios of stadard deviatios ad meas of the series are as follows u u = =. v v = = SD of u = u = u (u ) = 657 (.) = 6. 9 = 7.8 SD of v = v = v (v) = = 8 (.) Sice SD is ivariat to chage of origi x = u = 7.67 ad y = v =.76 CV of x = CV of y = x = 4 + u =4. = 7.9 y = 4 + v = 4. = = 7. 9 x x = 5.68 % y.76 = y 9. 9 = 8.4 % Sice the salesma A has smaller CV, he is more cosistet. Measures of Dispersio / 7

212 4. QUESTIONS FOR SELF - STUDY. What is dispersio? Why is it ecessary to take ito cosideratio the dispersio of. the data?. Defie rage as the measure of dispersio. Discuss its advatages ad limitatios. Metio some uses of rage.. Defie stadard deviatio. Establish its importace as a measure of variability? 4. What are the measures of absolute dispersio? Ca they be used for compariso of variability? 5. What are the relative measures of dispersio? I what respect are they superior to the absolute measures? 6. Defie coefficiet of variatio. I what situatios is it useful? 7. A set of observatios has the sum of squares of diviatio from the mea equal to. Fid its variace. If two more values, each equal to mea, are added, what will the variace of the ew set? 8. If all the observatios i the data are of same value, what will be its SD? 9. If x i (i =,...) are observatios o X, show that x ( i x i ). For observatios o Y, y = 5. Show that the mea of the data caot exceed 5.. Are the data =, x = 5. x = 8, cosistet? Give reasos for your aswer.. A variable takes values,, Fid the mea ad variace.. From the followig distributio of milk co operative societies accordig to procuremet of milk per day (i liters), compute stadard deviatio Quatity of Milk Societies : A survey coducted to determie the distace travelled (i Km) per litre of petrol by ewly itroduced moped yielded the followig distributio. Distace No. of Moped Fid the stadard deviatio. 5. The polythee bags were supplied by two suppliers A ad B. These bags were tested for burstig pressure ad the followig data were obtaied. Pressure i Kg. : Bags A : Bags B : Which supplier's bags have more cosistecy i burstig pressure? 6. Marks obtaied by two studets i the te differet papers at a examiatios are give below. Fid who is more cosistet. Mathematics & Statistics / 8

213 Marks A : Marks B: The mea of 5 observatios is 4.4 ad the variace is 8.4. If three of the five observatios are, ad 6, fid the other two. 8. The statistics of rus scored by the batsme A ad B i o iigs are give below. Player A Player B Mea 5 45 Stadard deviatio 4 6 Which of the two players is more cosistet? 9. Fid the stadard deviatio of the followig frequecy distributio. x : 4 5 f : k k k 4k 5k. For a group of observatios o X x = 45 ad x = 47. Fid the stadard deviatio. 4. SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi. Pre-degree Mathematics by Vaze, Gosavi Measures of Dispersio / 9

214 NOTES Mathematics & Statistics /

215 Chapter 5 Correlatio 5. Objectives 5. Itroductio 5. Correlatio 5.. Positive & Negative Correlatio 5. Covariace 5.4 Coefficiet of Correlatio 5.4. Properties of Correlatio Coefficiet 5.4. Iterpretatio of the value of Correlatio Coefficiet 5.4. Computig Correlatio Coefficiet For Ugrouped Data 5.5 Summary 5.6 Check Your Progress - Aswers 5.7 Illustrative Examples 5.8 Questios for Self - Study 5.9 Suggested Readigs 5. OBJECTIVES After studyig this chapter you will be able to explai followig - Two variables Relatios betwee two variables Positive relatios Negative relatio Reduce the egative relatio Bivariate Data : 5. INTRODUCTION The data we have studied upto this stage were cosistig of observatios o a sigle variable ad are called the uivariate data. But there are may situatios i which we are iterested i observatios o two variables for every uit i a sample or a group of uits. If we observe cosumptio of coal X ad productio of electricity Y for days i a moth. We get pairs of values (x i, y i ) for i =,,... These data o two variables costitute bivariate data. I short the set of pairs of observatios o two variables are called bivariate data. For example, the observatios o: (i) Age of husbad ad age of wife i several married couples. (ii) (iii) Itelligece quotiet ad score i aptitude test of studets i a class. Supply ad price of a commodity i a market o several days, are some examples of the bivariate data. 5. CORRELATION The major iterest i collectio ad study of bivariate data is i fidig whether there is ay mutual relatio betwee the two variables uder cosideratio or ot. This mutual or joit relatio betwee the two variables is called correlatio which ca be ascertaied by studyig the joit variatio of the two variables i the Correlatio /

216 data. For example, if we observe the data o cosumptio of coal ad productio of electricity at a thermal power plat, we fid that there is relatio betwee these variables because more cosumptio of coal is boud to produce more electricity ad shortage of coal is boud to result i shortage of electricity produced. I fact cosumptio of coal is the cause of productio of electricity. Uless there exists such a logical relatioship betwee the two variables the study of correlatio will be meaigless. There is o poit i studyig correlatio betwee height ad itelligece quotiet of a group of adults. Thus two variables are said to be correlated whe chage i value of oe variable causes correspodig chage i the value of the other variable. Populatio of a tow ad umber of vehicles i the tow are correlated because tows with larger populatio are boud to have larger umber of vehicles. 5.. Positive ad Negative Correlatio : The Variables showig correspodig chages i their values are said to be correlated. But these chages i differet pairs of variables are ot of the same kid. I some cases the chages i values of both the variables are i the same directio. Icrease i value of oe variable causes icrease i value of the other variable. Correlatio betwee these variables is said to be positive. The cosumptio of coal ad the amout of electricity produced are positively correlated. I some other cases the chages i the values of the two variables may be i opposite directio. Icrease i value of oe variable may cause decrease i value of the other variable. These variables are said to be egatively correlated. Sice ample supply of a commodity results i fall of price ad scarcity results i rise of price the variables supply of a commodity ad its price have egative correlatio betwee them. 5. COVARIANCE As stated above, the study of correlatio betwee two variables is i some sese a study of joit variatio of the two variables which may be termed as covariatio. I order to ascertai the degree of correlatio we eed a measure of this degree of covariatio. Such a measure is provided by covariace which is defied as the mea of products of deviatios of the observed values of X ad Y from their respective meas. Let us have a sample of pairs of observatios (x i y i ) o the variables X ad Y. The the meas of X ad Y for the give sample are x x i y ad y The the covariace of X ad Y for the give sample is x i xy i y Cov. (x, y) = x = i y i x y i The iterestig feature of this measure is that it may be egative, zero or positive accordig to the ature of correlatio betwee the variables. I case of data o positively correlated variables the covariace is also positive. Let us ow study the effect of chage of origi ad scale. Let us chage the variables X ad Y ito u ad v by the trasformatio. u = X a Y a ad v = h k The x i = a + hu i ad y i, = b + kv i, x = a + h u ad y = b + k v Mathematics & Statistics /

217 x i x = h(u i u ) ad y i y = k(v i v ) Hece Cov. (X, Y) = ( x i x) (yi y) = hk ( u i u) (vi = hk Cov. (u, v) v) Thus covariace is ivariat to the chage of origi but ot to the chage of scale. 5.4 COEFFICIENT OF CORRELATION For further study of correlatio ad compariso of correlatio it is ecessary to measure the degree of correlatio betwee the two variables uder cosideratio. Professor Karl Pearso has suggested a measure of a degree of correlatio called coefficiet of correlatio. Karl Pearso's Coefficiet: It is defied as the ratio of covariace of two variables to the product of stadard deviatios of these variables. It is also kow as product momet correlatio coefficiet. The coefficiet of correlatio computed for a sample from a bivariate populatio is deoted by r. Let us have a sample of pairs of observatios (x i, y i ) o variables X ad Y. The the sample correlatio coefficiet is give by Cov.(x, y) r = x y Sice Cov. (X, Y) = x i yi x y ad x = x i x ; y = O simplificatio we get r = x i y x y i y i ( xi x ) ( y i y y This form is more suitable for computatio of correlatio coefficiet. The Magitude or umerical value of r measures the degree of correlatio ad the algebraic sig of r idicates the type of correlatio positive or egative. Thus the value of correlatio coefficiet gives us the complete idea about the correlatio betwee two variables Properties of Correlatio Coefficiet: Let r be the coefficiet of correlatio betwee the variables X ad Y computed from the sample of pairs (x i y i ) (i) The magitude of the coefficiet of correlatio i.e r is ivariat to the chage of origi ad scale. Let us deote the coefficiet of correlatio betwee X ad Y by r xy ad that betwee u ad v by r uv Let us chage the variables X ad Y ito u ad v by the trasformatio. u = x a y b ad v = h k The. X = a + hu ad Y = b + kv ad r xy = r uv This shows that the umerical value of the correlatio coefficiet is ivariat to the chage of origi ad scale. Correlatio /

218 Further it ca be cocluded that (i) whe h ad k have same algebraic sig i.e. whe both are positive or both are egative, r xy = r uv ad (ii) whe h ad k have differet algebraic sigs r xy = r uv. For example, if coefficiet of correlatio betwee X ad Y is.8 that betwee X ad Y is.8. Also the coefficiet of correlatio betwee X ad Y is.8. But the correlatio betwee X ad Y or that betwee X ad Y is.8. Further the coefficiet of correlatio betwee (X + 5) ad (Y ) is the same as r xy but that betwee (X + 5) ad ( y +) is r xy ii) Karl Pearso's coefficiet of correlatio betwee two variables is umerically less tha or equal to uity Iterpretatio of the value of Correlatio Coefficiet: The umerical value of the correlatio coefficiet measures the degree of correlatio betwee the two variables. The larger value of r idicates closer relatioship betwee the variables. Whe r >.8, it idicates correlatio of very high degree. Whe r lies betwee. ad.7, oe ca say that there is sigificat or cosiderable, correlatio betwee the two variables. Correlatio is said to be very poor or isigificat whe r <.. The algebraic sig of r idicates whether the correlatio is positive or egative. The value r = meas that the variables are ucorrelated. Whe r = or r =, there is perfect positive correlatio or perfect egative correlatio respectively, betwee the two variables. But these values of r are very ucommo. I real life situatios, chace of occurrece of these values of r are very rare. I all these iterpretatios it is assumed that the sample from which r is computed is sufficietly large Computig Correlatio Coefficiet For Ugrouped Data : The data specifyig all the pairs of observatio (x i, y i ), i =,... ; o two variables X ad Y are called ugrouped data. The steps i computig correlatio coefficiet for these data are give below : i) Compute meas of X ad Y ii) x x i ad y y Compute stadard deviatios of x ad y i i x = x (x), y = y i (y) This eeds computatio of sums of squares x i ad y i iii) Compute the sum of products x i y i. xi y The Cov. (x, y) = i x y. iv) Compute the coefficiet of correlatio betwee X ad Y. Cov.( x, y) r = x y Note : If the values of X ad Y i the data are icoveietly large so as to make computatio of sums of squares ad sum of products difficult, we may employ the techiques of chage of origi ad/or chage of scale that would simplify the computatios : Usually the scalig factors are both positive. So the value of r remais ualtered. For example, if we have the trasformatio u = X h a Y ad v = a k where h >, k >, r xy = r uv Mathematics & Statistics / 4

219 Check Your Progress to 5.4. What is correlatio?. What is positive correlatio & egative correlatio?. What is Covariace? 4. What is Karl Pearso's coefficiet of correlatio? 5. Choose the correct aswer from give. (i) If X ad Y are ay two radom variables the the covariace betwee ax + b, cy + d is give by a) cov(x, Y) b) abcd cov(x, Y) c) ac cov(x, Y) d) ac cov(x,y) + ab (ii) The correlatio coefficiet betwee college etrace exam grades ad the fial gardes was computed to be.8. O the basis of this you would recommed that: a) the etrace exam is a good predictor of success b) studets who do worst i this exam will do best i fial c) Studets at this school are ot scholars d) Recomputed the correlatio coefficiet (iii) The correlatio coefficiet betwee X ad Y is kow to be zero, We the coclude that a) X ad Y have stadard distributios. b) the variaces of X ad Y are equal. c) there exists o relatioship betwee X ad Y d) there exists o liear relatioship betwee X ad Y (iv) Suppose the correlatio coefficiet betwee height as measured i feet ad weight as measured i pouds is.4. What is the correlatio coefficiet of height measured i iches versus weight measured i ouces ( iches = oe feet, 6 ouces = oe poud) a).4 b). c).5 d) caot be determied from the iformatio give (v) Cosider the followig data. x 4 y which oe of the followig would be true? (a) Correlatio coefficiet betwee X ad Y is egative but ot equal to. (b) Correlatio coefficiet betwee X ad Y is. (c) Correlatio coefficiet betwee X ad Y is. (d) Noe of the above. Correlatio / 5

220 5.5 SUMMARY The mutual or joit relatio betwee the two variables is call correlatio. The two variables are said to be correlated whe chage i value of oe variable causes correspodig chage i the value of the other variable I some cases the chages i values of both the variables are i the same directio. Icrease i value of oe variable causes icrease i value of the other variable such a variables are called as positively correlated variables. I some other cases the chages i the values of the two variables may be i opposite directio. Icrease i value of oe variable may cause decrease i value of the other variable. These variables are said to be egatively correlated. The study of correlatio betwee two variables termed as covariatio. A measure of the degree of covariatio is called as covariace. Accordig to professor Karl Pearso a measure of a degree of correlatio called coefficiet of correlatio. It is defied as the ratio of covariace of two variables to the product of stadard deviatios of these variables. It is also kow as product momet correlatio coefficiet. 5. to CHECK YOUR PROGRESS ANSWERS. The mutual or joit relatio betwee the two variables is called correlatio.. Icrease i value of oe variable causes icrease i value of the other variable such a variables are called as positively correlated variables. Icrease i value of oe variable may cause decrease i value of the other variable. These variables are said to be egatively correlated.. The study of correlatio betwee two variables termed as covariatio. A measure of the degree of covariatio is called as covariace. 4. Accordig to Professor Karl Pearso a measure of a degree of correlatio called coefficiet of correlatio. It is defied as the ratio of covariace of two variables to the product of stadard deviatios of these variables. It is also kow as product momet correlatio coefficiet. 5. (i) c, (ii) d, (iii) d (iv) a (v) a 5.7 ILLUSTRATIVE EXAMPLE Example : The followig are the values of exports of raw cotto (X) ad the values of imports of maufactured cotto goods (Y) i Crores of Rs. Compute the coefficiet betwee X ad Y. Table 5. : Computatio of Coefficiet of Correlatio X Y y = u 7 v = Y 6 uv u v Mathematics & Statistics / 6

221 Here the give values of X ad Y are large. So we covert X ad Y ito u ad v by chage of origi. Take u = X 7 ad v= Y 6 The u i 8 u = v 5 v = = The stadard deviatios are u i u = (u ) = 99 7 ( 5.485) = 9.94 v = v v i u v Cov (u,v)= i i = = u v 6 = ( 5.485) (.74) = = Cov (u, v) Hece r uv = = x8.65 u v Sice the coefficiet of correlatio is ivariat to chage of origi we have r xy = r v =.94 This shows that there is correlatio of high degree betwee the variables X ad Y. Note : Trasformatio of variables eed ot be used uless it sigificatly facilities computatios. Mere it is used oly as a illustratio. 5.8 QUESTIONS FOR SELF - STUDY. Explai with a example the cocept of bivariate data.. Whe are two variables said to be correlated? What do you mea by (i) positive correlatio ad (ii) egative correlatio? Give two examples of each type.. Defie Karl Pearsio correlatio coefficiet ad state its properties. 4. Show that the correlatio coefficiet is umerically ivariat to the chage of scale ad origi. 5. Te pairs of values of X ad Y give the followig result: x = 4, y = 5, x =, y = 5 ad xy = 6. Fid the correlatio coefficiet betwee X ad Y. 6. Twety pairs of values of X ad Y give x = 5, y =, Ex = 68, y = 5 ad xi (yi y) =. Fid the coefficiet of correlatio 7. From the data of 5 pairs of observatios of X ad Y a studet got x =, y = 5, y =, xy = 5. Are these result cosistet? 8. Two series of X ad Y with 5 observatios have stadard deviatios 4.5 ad.5 respectively. The sum of products of deviatios of X ad Y from their respective meas is 4.. Fid the coefficiet of correlatio betwee X ad Y. 9. From the followig data of supply i quitals (X) ad price i Rs. per quital (Y) of a certai commodity compute the correlatio coefficiet betwee price ad supply. Correlatio / 7

222 X : Y: From the followig data of height X i cm ad weight Y i kg. of adults fid the correlatio coefficiet betwee X ad Y. X: Y : The mea soil temperature (X) ad umber of days (Y) required for germiatio for witer wheat at places are give below : X Y: Compute the correlatio coefficiet betwee X ad Y.. From the followig data o water X (i ft) ad yield of Alfalfa Y (i tos per acre) calculate the correlatio coefficiet betwee X ad Y X: Y: From the data of pairs of observatios of X ad Y followig result are obtaied x = 99, y = 94, (X x) = 98, (y y) = 6 ad (X y) = 6. Fid the coefficiet of correlatio. 4. Fid, if r =.5, y = 8, (X j x) = 9 ad (x i x) (y i y) =. 5. Give =, = 8, y = 4, x = 68, y =, xy= 48, fid the correlatio coefficiet betwee x ad y. 6. Compute the correlatio coefficiet betwee x ad y from the followig : =,, x =, y = 5, (x ) = 8, (y 5) = 5, (x ) (y 5) = 6 7. Give r xy =.75, fid the correlatio coefficiet betwee a) (x ) ad (y 5) b) (x 4)ad( y) c) x ad 5 y 5.9 SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi. Pre-degree Mathematics by Vaze, Gosavi Mathematics & Statistics / 8

223 NOTES Correlatio / 9

224 NOTES Mathematics & Statistics /

225 Chapter 6 Liear Regressio 6. Objectives 6. Itroductio 6. Lie of Regressio 6. Equatio of Lie of Regressio by the Method of Least Squares 6.4 Iterpretatio of Coefficiet of Regressio 6.5 Properties of Coefficiet of Regressio 6.6 Summary 6.7 Check Your Progress - Aswers 6.8 Illustrative Examples 6.9 Questios for Self Study 6. Suggested Readigs 6. OBJECTIVES After studyig this chapter you will be able to discuss followig Regressio Two variable Three variables Quatitative evidece Sophisticated Results Proper equatio Iterpretatio of results Use of results (decisios) 6. INTRODUCTION I the precedig chapter we have studied methods of measurig the, degree of correlatio betwee the two variables by obtaiig bivariate data o these variables. If the bivariate data provide a quatitative evidece of existece of correlatio or associatio betwee the variables, our attempt would be to establish this associatio i some fuctioal form mathematically, that would eable us to estimate quite accurately, o a average, the value of oe variable o the basis of the value of other variable. Such a mathematical relatioship betwee two variables is called regressio equatio or simply regressio. This estimatio by associatio is quite sophisticated ad very useful. This procedure is actually that of predictio ad predictio is the cetral fuctio of scieces. The mai task of ay scietific study is to discover the geeral relatioships betwee the observed variables ad to state the ature of such relatioships i mathematical terms, so that the value of oe variable ca be predicted o the basis of that of aother. This is what we are goig to attempt i this chapter. Geerally, the relatioships betwee the variables uderstudy such as i) height ad weight of adult me ii) umber of ifat deaths ad umber of births etc. are very blurred, vague ad imprecise. Ordiary mathematical methods are ot useful i this case but statistical methods are. The special cotributio of statistics i this field is that of hadlig such vague, blurred, ad imprecise relatioships.' Liear Regressio /

226 As stated above the mathematical relatioship betwee the two variables uder study is called regressio equatio which is essetially a predictio equatio. But the term regressio is well established i statistics ad o attempt has bee made to replace it. 6. LINE OF REGRESSION The simplest equatio for expressig the relatioship betwee the two variables is liear equatio. I the case the regressio is kow as lier ad the equatio is called the lie of regressio. Amog the two variables uder cosideratio the regressio equatio expresses oe variable i terms of the other. If the equatio expresses Y i terms of X, Y is called 'depedet' or 'explaied' variable ad X the 'idepedet' or 'explaatory' variable. [Note that the term 'idepedet' is ot used i statistical sese.] Thus the equatio Y = a + bx is called lie of regressio of Y o X ad is used for predictio of Y for give X. Here a ad b are costats for the give lie. The coefficiet b of X, is called the regressio coefficiet of Y o X. Likewise the equatio X = a' + b'y gives the lie of regressio of X o Y ad is used for predictio of X for give Y. The coefficiet b' is the regressio coefficiet of X o Y. There is oly oe measure of degree of correlatio betwee the two variables X ad Y.it is the correlatio coefficiet r. But for the same pair of variables we have two lies of regressio because we have two choices for depedet ad idepedet variables. The coefficiet of correlatio r xy is ot differet from r yx. Hece there is oly oe coefficiet of correlatio for the give pair of variables. The costats i the regressio equatio are determied that fits the data is obtaied by the priciple of least squares. 6. EQUATION OF LINE OF REGRESSION BY THE METHOD OF LEAST SQUARES Let us have a sample of pairs of observatios (x i, y i ) o the variables X ad Y. Let the equatio of lie of regressio of Y o X be Y = a + bx..() For the i th observatio, y i is the observed of Y. The value of Y obtaied from the equatio () for X = X i, is called the liear regressio estimate of Y deoted. y i Thus y i = a+bx i.... () Now the costats a ad b i the equatio () are evaluated so that the sum of squares of deviatios of observed y i from their regressio estimates This is kow as the method of least squares. y i is the least. Let the sample of pairs (x i y i ) have the meas x ad y ad the variaces X ad y for X ad Y respectively ad let Cov (X, Y) = m be covariace betwee X ad Y for the sample. Let D = (y i y i form the liear regressio estimate ), which is the sum of squares of deviatios of observed y i y i The costats a ad b are foud i such a way that D is miimum. These values of a ad b will be give by the equatios. a + b x j = y i.... () ax i + bx i = x i y i... (4) Mathematics & Statistics /

227 The sums X i, y i, x i ad X i y i are kow from the data. Thus we have two equatios ad 4 i two ukows a ad b. We ca obtai a ad b by solvig these equatios for a ad b. The equatios () ad (4) are called ormal equatios. From equatio (4) we have y i a= b i = x y b x substitutig this value of a i (4), we get ( y b x ) x i + b x i = x i y i. Now x i = x This gives ( y b x ) ( x ) + bx i = x i y i b= Cov(x, y) x m = x Substitutig these values i () the equatio of lie of regressio is writte as Y = y b x + bx or Y = y + b(x x )...(5) Likewise the equatio of lie of regressio of X o Y obtaied by the method of least squares is X = x + b'(y y )...(6) Cov( x, y) Where b = y From the equatios of lies of regressio as give i (5) ad (6) it is evidet that both the lies pass through the poit ( x, y ). Thus ( x, y ) is the poit of itersectio of the two lies of regressio. The lies of regressio also ca be expressed as (Y y ) = b(x x ) which is the lie of regressio of Y o X ad (X x ) = b'(y y ) which is the lie of regressio of X o Y. I this form the equatios are easy to memorize. 6.4 INTERPRETATION OF COEFFICIENT OF REGRESSION Cosider lie of regressio of Y o X. I this form the equatio is of the form Y = a + bx. Here b is the coefficiet of regressio of Y o X. From the equatio of lie of regressio it is clear that for uit chage i value of X, the value of Y will chage by b uits. This b is the rate of chage of value of Y for uit chage i X. If b is positive the icrease i value of X will be associated with icrease i value of Y i.e. the correlatio betwee X ad Y will be positive. O the cotrary if b is egative icrease i value of X will correspod to decrease i value of Y, showig that there is egative correlatio betwee X ad Y. I geeral the coefficiet of regressio gives the rate of chage of depedet variable per uit chage i value of idepedet variable ad the algebraic sig of the coefficiet of regressio determies whether the correlatio is positive or egative. Liear Regressio /

228 6.5 PROPERTIES OF COEFFICIENT OF REGRESSION Let a sample of pairs of values (x i y i ) of the variables X ad Y give the variaces x ad y ad the coefficiet of correlatio r. Let b ad b' be the coefficiets of regressio of Y o X ad X o Y respectively, The we kow that Cov. (X, Y) = r x y ad the regressio coefficiets b ad b' are give by Cov(X, Y) Cov(X, Y) b = ad b = x y Cov (X, Y) The b= x Cov (X, Y) ad b = y r x y y =r x x r x y = x y r = = a) Sice x ad y are always positive we ca say that both the regressio coefficiets have the same algebraic sig which is the same as that of correlatio coefficiet. Also from the values of b ad b' it follows that bb' = r y r x x y = r The product of regressio coefficiets is equal to r. Thus the values of b ad b will be said to be cosistet if (i) both have the same algebraic sig ad (ii) their product is less tha of equal to uity, as r <. b) The regressio coefficiet is ivariat to the chage of origi but ot to the chage of scale. Sice every regressio coefficiet is a ratio of covariace to variace it is ivariat to the chage of origi. The reaso for this is that both the covariace ad the variace are cetral momet which are kow to be ivariat to the chage of origi. Let us have u = h X ad v = k Y The x = h u, ad y = k v ad Cov. (X, Y) = hk Cov. (U, V) Let b vu be the coefficiet of regressio of v o u. Cov ( u, v) The bvu = ad the coefficiet of regressio of Y o X is the give by u Cov.( X, Y ) b = byx = = x u y hk Cov.(u, v) h = h k bvu ad b = b xy= k h buv From this it follows that if the scalig factors h ad k for the two trasformatio are equal the regressio coefficiet will be ualtered. Mathematics & Statistics / 4

229 Check Your Progress to 6.5. What is Lie of Regressio?. What is Regressio Co-efficiet?. Choose the correct aswer from the give. i) Based o the data {(x i, y i ), i =, } the two regressio lies are y = /5 + /5 x ad x = /5 + /5x. Let m x, m y deote sample meas. ii) (a) The two lies actually collapse ito oe lie ad the correlatio coefficiet is. (b) correlatio coefficiet is / (c) m x = m y = (d) m x = m y = / Liear regressio of Y o X is Y = + X. Correlatio coefficiet betwee y ad x is /. The the regressio coefficiet b x,y of x o y is; (a) (b) ½ (c) /4 (d) / 6.6 SUMMARY The mathematical relatioship betwee the two variables uder study is called regressio equatio which is essetially a predictio equatio. The simplest equatio for expressig the relatioship betwee the two variables is liear equatio. I the case the regressio is kow as lier ad the equatio is called the lie of regressio. The equatio Y = a + bx is called lie of regressio of Y o X ad is used for predictio of Y for give X. Here a ad b are costats for the give lie. The coefficiet b of X, is called the regressio coefficiet of Y o X. 6. to CHECK YOUR PROGRESS - ANSWERS. The simplest equatio for expressig the relatioship betwee the two variables is liear equatio. I the case the regressio is kow as lier ad the equatio is called the lie of regressio. The equatio Y = a + bx is called lie of regressio of Y o X ad is used for predictio of Y for give X. Here a ad b are costats for the give lie. The coefficiet b of X, is called the regressio coefficiet of Y o X. (i) d (ii) c 6.8 ILLUSTRATIVE EXAMPLES Example : I a agricultural experimet o the study of effect of depth of water i the soil (X) i ft. o the yield of as crop i b. per plot (Y) the followig data were obtaied. X: Y: Obtai the equatio of lie of regressio of Y o X ad estimate the yield whe the depth of water i the soil is ft. Liear Regressio / 5

230 Solutio : The Steps i computatios are as follows : i) Fid the sum of squares ad products. ii) iii) x i =. y i = 9 x i = 6.45 y i =75 Xi; y i = 47 The umber of observatios = 7. Compute the meas of X ad Y x =. 9 =.9 ad y = = Compute the variace of X x = x i x 6.45 = (.9) 7 = =.685 iv) Compute the covariace x y Cov (X,Y) = i i x y 47 = (.9) (98.57) 7 = = 5.86 v) Let the equatio of lie of regressio of Y o X be Y = a + bx The b = = Cov.(X,Y) x =.85 a = y b x ] = (.85) (.9) = Hece the equatio of lie of regressio is Y = The regressio estimate of Y whe X = is Y = 8.76 Thus the liear regressio estimate of yield of crop whe the depth of water is ft., is 8.76 Ib. Example : The followig table shows the meas ad the stadard deviatios of prices of shares of two compaies. Mathematics & Statistics / 6

231 Compay Mea Price Stadard deviatio A Rs. 9.5 Rs..8 B Rs Rs. 6.8 The coefficiet of correlatio betwee the prices of two shares is.4. Fid the most likely price of shares of compay A whe the price of share of compay B is Rs. 55. Solutio : Let the prices of share of compay A ad compay B i Rs. be X ad Y respectively. The we are give that the meas of X ad Y are x = 9.5 ad y = 47.5 ad the stadard deviatios are x =.8 ad y = 6.8. Also the coefficiet of correlatio is r =.4. Now to estimate the price of shares of compay. A i.e. value of X we are give the price of shares of compay B i.e. Y = 55. For this we have to use the equatio of lie of regressio of X o Y. Let this equatio be X = a + b Y. Here b=r x y =.4 = ad a = x b y of X. = a = = 6.68 The equatio of lie of regressio is X = Y The most likely price of shares of compay A is the liear regressio estimate For Y = 55 this estimate of X is give by x = = 4.5 The most likely price of shares of compay A is Rs. 4.5 whe that of compay B is Rs. 55. Example : Give the two liear regressio equatio 8X + Y + 66 = ad 4X 8Y = 4 ad V (X) = 9, fid the meas of X ad Y, the correlatio coefficiet betwee X ad Y ad V(Y). Solutio : We kow that the coordiates of poit of itersectio of the two lies of regressio are x ad y, the meas of X ad Y. The regressio equatios are 8X - Y = 66 ad 4X 8Y = 4 Solvig these equatios we get X = ad Y = 7. Hece the meas of X ad Y are x = ad y = 7. Now to fid the correlatio coefficiet we have to fid the regressio coefficiets b ad b'. For this we have to choose oe of the lies as that of regressio of Y o X ad Liear Regressio / 7

232 the other is the the lie of regressio of X o Y. Let 8X - Y + 66 = be the lie of regressio of Y o X. This gives Y = 8 x =.8x The coefficiet of X i this equatio is b =.8. The the other equatio is that of lie of regressio of X o Y which ca be writte as 8 4 X = Y Here the regressio coefficiet b' = =.45 4 Now r = b b' =.8.45 =.6 r = ±.6 The correlatio coefficiet has the algebraic sig same as that of b r =.6 [Note : We choose arbitrarily the lies as that of regressio of Y o X or X o Y. If the product b b' is less tha uity, our choice is correct. Otherwise we have to take the other chose. Fortuately there are oly two choices]. Now the coefficiet of regressio of Y o X is r y x y b = r =.6 =.8 y x y =.8 = 4, Hece V (Y) = QUESTIONS FOR SELF - STUDY. Explai the cocept of regressio ad its utility.. Why do we have two lies of regressio (i) of Y o X ad (ii) X o Y?. What do you mea by a liear regressio coefficiet of Y o X? How will you iterpret the value of it? 4. If b yx ad b xy are the coefficiets of regressio of Y o X ad X o Y respectively, show that b yx, b xy = r 5. Brig out the icosistecy, if ay, i the followig : i) b =.6, b' =.5 ii) b =., b' =.5 iii) b = b =.5 ad r =.7 5. The followig table gives the ifat mortality rate (X) ad birth rate (Y) for eight years. X: Y: Obtai the lie of regressio of birth rate o ifat mortality rate ad estimate the birth rate for the ifat mortality rate The followig data give live weight of a pig (X) ad weight of a side of baco (Y) X : Y : Mathematics & Statistics / 8

233 Estimate the lie of regressio of weight of pig o weight of a side of baco ad calculate the weight of pig if the weight of baco is. 7. The umber of defective items produced per uit time, Y, by a certai machie is thought to vary directly with the speed of the machie, X measured i r p.m. Observatios for hours selected at radom from a moth give the followig results. X: Y: Estimate the lie of regressio of Y o X ad the umber of defectives per hour whe the speed of the machie is r.p.m. 8. The followig radom sample gives the umber of hours of study (X) for a examiatio ad the grades Y obtaied by studets. X: Y: Obtai the lie of regressio of grades o hours of study. 9. The average price of shares was Rs. 5 ad the average gai per share was Rs. 7. The coefficiet of regressio of gai per share (Y) o the price (X) was.5, Estimate the gai per share for the price Rs... Twelve observatios o the price (X) of shares ad the volume of sales (Y) at Bombay stock exchage gave the followig results. x = 58, y = 7, xy = 494, x = 4568 ad y = 76. Obtai the equatio of lie of regressio of volume of sales o price of shares. Predict the volume of sales (i thousads of shares) for shares of price Rs. 4/.. Give the followig data, obtai the liear regressio estimate of X for Y =. xi = 7.6, yi = 4.8, x =.6, y = 5, r =.8. The two regressio lies are x y = ad 4y 5x + 7 =. Fid the meas of X ad Y. If stadard deviatio of X is, fid that of Y.. Fid the meas of X ad Y ad the correlatio betwee X ad Y, if the equatios of lies of regressio are y x 5 = ad y x =. 4. The equatios of two lies of regressio are X + Y 6 = ad 6X + Y = Fid the meas of X ad Y. Estimate Y for X =. 5. Give the meas of X ad Y, 5 ad. The lie of regressio of Y o X is parallel to the lie Y = 9X + 4 ad correlatio coefficiet is.6. Estimate the value of X whe Y =. 6. I the regressio aalysis of a problem the equatios of lies of regressio were foud to be X 4Y = 8 ad Y 9X + 4 =. The variace of Y was 6. Fid the meas of X ad Y, x ad the coefficiet of correlatio. 6. SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi. Pre-degree Mathematics by Vaze, Gosavi Liear Regressio / 9

234 NOTES Mathematics & Statistics /

235 Chapter 7 Idex Numbers 7. Objectives 7. Itroductio 7. Uses of Idex Numbers 7. Price Idex Numbers 7.4 Problems i Costructio of a Idex Number 7.5 Summary 7.6 Check Your Progress - Aswers 7.7 Questios for Self - Study 7.8 Suggested Readigs 7. OBJECTIVES After studyig this chapter you will be able to calculate ad explai- Idex umbers Types of idex umbers Costructio of Idex umbers Uses of Idex umbers 7. INTRODUCTION May a time we are iterested i kowig the relative chages i values of variables like populatio, prices, idustrial productio, agricultural productio, exports, imports etc. over a period. Oe of the ways of measurig these chages is the idex umbers. We are quite familiar with the wholesale price idex, cosumer's price idex, Bombay stock exchage idex which give us the kowledge of the degree of chages i correspodig variables. A idex umber ca be defied as the device used for measurig the relative chages i value of a variable or of a group of related variables from oe period to other or from oe place to aother place. 7. USES OF INDEX NUMBERS The price ad quatity (of cosumptio) idex umbers have i recet years become he importat tools i iterpretatio of the ecoomic coditios of a state. Rapid ad erect chages i price idex or idex of price of shares idicate ustable ecoomic coditios. Whereas stability i these idex umbers idicate stable ecoomy. I that sese these idex umbers are called Ecoomic Barometers. Wages of employees are closely tied with the cosumer's price idex of that locality. Revisio of pay scales, fixatio of dearess allowaces, miimum wages, pesio policies are liked with price idex umbers. So the price idex umbers are closely watched by the employers as well as employees. The idex of idustrial productio is of great iterest to busiessme ad the studets of atioal ecoomy because it furishes the iformatio o the curret positio of atioal productio. The price idex umbers are also useful i determiig the purchasig power of moey. It ca also be used for determiig the real icome or real wages of employees. The idex umbers like itelligece quotiet are useful i Psychology ad Educatio. The populatio idex is of iterest to the studets of sociology ad demography. Although there ca exist may idex umbers for differet purposes a commo ma is more cocered with the price idex umbers. So for the further discussio let us restrict ourselves to the price idex umbers oly. Idex Numbers /

236 7. PRICE INDEX NUMBERS For measurig the relative chage i price of a sigle commodity or a group of commodities we use price idex umber. Geerally we are iterested i measurig chages i prices over a chage of time. The chage is measured from some fixed period of referece kow as base period The period which is compared with this base period is called curret period. Let us use the otatio p ad p for the prices i base period ad curret period respectively. The we use the ratio of prices for measurig the chage because the ratio is idepedet of uits i which the prices are expressed. The relative importace of differet commodities ca be cosidered by usig weights. The weights used are proportioal to the quatities of cosumptio or the value of goods ad services i the series. Such idex umbers are called weighted Idex umbers. Differet systems of weights used give rise to differet formula. Some of these are give below. (a) Laspeyre's Idex: This idex umber is the ratio of weighted aggregate prices usig the quatities cosumed i base year (q ) as the weights. P = p q p q. This ca be iterpreted as the ratio of the value of basket of goods cosumed i base year accordig to curret year ad base year prices. This idex umber always gives a upward bias. (b) Paasche's Idex : This idex umber is also a ratio of weighted agreegate of prices usig the quatities cosumed i curret year as weights pq P = p q This idex ca be iterpreted as ratio of value of good cosumed i curret year to prices i curret year ad base year. This idex as cotrary to Laspeyre's Idex, is foud to give a dowward bias, (c) Fisher's Idex: Sice either Laspeyers formula or Paasche's formula give a correct idea of chage i price level, Irvig Fisher suggested that the geometric mea of these two idex umbers will give the suitable idex umber. Accordig to him the idex umber is give by P = p q p q pq p q This gives a more accurate price idex but it lacks i iterpretatio. However, it has may desirable properties ad hece it is kow as Fisher's Ideal Idex Number. Example : Calculate the Laspeyre's, Paasche's ad Fisher's idex umbers for prices i the year 987 with 98 as base year from the followig data. Commodity Base year (98) Curret Year (987) Price (P o ) Quatity (q o ) Price (P ) Quatity (q ) Rice Wheat Jawar Pulses 6 8 Mathematics & Statistics /

237 Solutio : Computatios for the required idex umbers are show i the followig table. Commodity P o Q o P q p o q o p q p q p q Rice Wheat Jawar Pulses TOTAL Laspeyre's Idex pq p = p q 47 = 4 =8.5 Paasche's Idex p q P a = p q 5 = 7 =5.5 Fisher s Idex P F = P P L a = 8.5x 5. 5 = PROBLEMS IN CONSTRUCTION OF AN INDEX NUMBER Costructio of ay idex umber itself a difficult task. It poses may problems i the process of costructio of a idex umber. Followig are the commo problems which eed a careful ad thoughtful cosideratio while costructig a idex umber. (i) Specificatio of the Purpose ad Scope of Idex Number : Every idex umber is costructed with some defiite purpose ad its uses are also limited. There does ot exist a all purpose idex umber. The wholesale price idex umber caot be used for comparig the retail price levels i two periods. The cosumer's price idex for textile workers caot be used for comparig cost of livig of higher icome group. So it is very importat ad ecessary to specify the purpose at the outset oly. It govers the further details of costructio of a idex umber. It also defies the proper use of idex umber uder cosideratio. (ii) Selectio of Items : Selectio of the items ad their umber is govered by the purpose itself. Oly relevat items should be icluded i the series which have direct ifluece o the idex umber. The umber of items to be icluded should be eough to make it represetative ad it should ot be too large also as it would create difficulties i collectig the price data ad iformatio o weights. The items should suit the tastes, habits ad customs of the class of people for whom the idex umber is costructed. Cosideratio should be give to the quality of the items like rice, wheat, recreatio as these differ from class to class. (iii) Selectio of Weights : Geerally, the idex umber is a weighted idex umbers as it is more Idex Numbers /

238 (iv) realistic. Weights allow differet items to ifluece the ides umber to differet extets. The weights should be proportioal to the relative importace of the item. The method i which we use ushc weights is called explicit weightig. I some cases, we may iclude i the series more varieties of the same item which is more importat ad less varieties of items of less importace, this is idirect or implicit weightig. I explicit weightig, we geerally use the quatity of cosumptio or the value of goods cosumed as weights For collectig these data we have to coduct a sample equiry. Selectio of Base Year: The usefuless of the idex umber depeds to some extet o the choice of base year. So proper care should be take i selectig the base year also. The base year should to be too distat i the past. The patter of cosumptio is likely to chage with the time. Some items may become out of use ad some ew items may come i to use. this may lose the comparability of the periods. This ecessitates revisio of base year from time to time. The other importat poit i choice of base year is that it must be the period of ecoomic stability. The evets like wars, famies, epidemics are likely to create erratic chages i prices which are boud to be temporary. They reflect the istability of the ecoomy ad these coditios do ot prevail for a log time. So it is ecessary that the base year chose must be a year of ecoomic stability. (v) Selectio of Sources of Price Data : After the commodities have bee selected it is ecessary to collect the data o prices of these commodities for costructig a price idex umber. This prices are collected from the markets or shops from which useal purchases are doe. The cocessios or discouts should ot be take ito cosideratio. The prices should be for those qualities of the commodity which are commoly cosumed by the class of people uder cosideratio. The price may be collected by ivitig quotatios from the reliable sources ad agets. To esure reliability the quotatios for some commodities may be ivited from two or more agets. The price data also ca be obtaied from published reports of official agecies. (vi) Selectio of Average or Formula : Sice we are iterested i costructig a sigle idex which will summaries the chages i values of umber of related variables we have to select the proper average that will serve the purpose. Amog the various averages the media ad the mode are out of cosideratio. Oly the arithmetic mea ad the geometric mea ca be used. Amog them the A. M. is easy to uderstad ad to compute also. So i may cases we used the weighted A. M. of price relatives for costructig the idex umber. I practice, Laspeyre's formula is widely used as it uses the base year quatities as weights. The data o base year quatities are easily available the those o curret year quatities. 7. to 7.4 Check Your Progress.. What is Idex umber?. What is Price idex umber?. List the commo problems which eed to be cosidered while costructig a idex umber. Mathematics & Statistics / 4

239 4. Choose the correct aswer from give list. i) Laspeyre's idex umber is give by the formula a) c) p p q p q p q q b) d) p p q p q p q q ii) Paasche's idex umber is give by the formula. a) c) p p q q p q p q b) d) p q p q p q p q 7.5 SUMMARY A Idex umber ca be defied as the device used for measurig the relative chages i value of a variable or of a group of related variables from oe period to other or from oe place to aother place. Idex umbers are also called as ecoomic barometers: For measurig the relative chage i price of a sigle commodity or a group of commodities we use price idex umber. Commo problems i costructio of a Idex umber are : (i) Specificatio of the Purpose ad Scope of Idex Number. (ii) Selectio of Items. (iii) Selectio of Weights. (iv) Selectio of Base Year. (v) Selectio of Sources of Price Data. (vi) Selectio of Average or Formula. 7.6 CHECK YOUR PROGRESS - ANSWERS 7. to 7.4. A Idex umber ca be defied as the device used for measurig the relative chages i value of a variable or of a group of related variables from oe period to other or from oe place to aother place. Idex umbers are also called as ecoomic barometers.. For measurig the relative chage i price of a sigle commodity or a group of commodities we use price idex umber.. Commo problems i costructio of a Idex umber are : (i) (ii) (iii) (iv) (v) Specificatio of the Purpose ad Scope of Idex Number: Selectio of Items Selectio of Weights Selectio of Base Year Selectio of Sources of Price Data Idex Numbers / 5

240 (vi) Selectio of Average or Formula 4. (i) c (ii) b 7.7 QUESTIONS FOR SELF - STUDY. Explai the meaig ad the utilig of Idex umbers.. State Laspeyre's, Paasche's ad Fisher's formulae of idex umbers ad metio their specialties.. Give iterpretatio of Laspeyre's ad Pasche's Idex umber of price: 4. Discuss various problems i costructio of a Idex umber. 5. Compute Laspeyre's, Passche's ad Fisher's Idex umber for price from the followig data. Commodity Base Year Curret Year Price Quatity Price Quatity Price A B 4 5 C D Calculate Laspeyre's ad Pasche's Idex umber for price from the followig data ad commet o your results. (a) (b) (c) Commodity p o q o p q A.5 8 B C 8 5. Commodity p o q o p q Commodity p o q o p q I 9 6 II 8 4 III Mathematics & Statistics / 6

241 7. Calculate appropriate Idex umber i each of the followig. (a) Commodity A B C D Po P q (b) Commodity I II III P 4 7 P 5 5 q SUGGESTED READINGS. Mathematics ad Statistics by M. L. Vaidya, M. K. Kelkar. Statistical Aalysis by S. P. Aze ad A. A. Afifi. Pre-degree Mathematics by Vaze, Gosavi Idex Numbers / 7

242 NOTES Mathematics & Statistics / 8