Review 1 of Math 727, Probability c Fall 2013 by Professor Yaozhong Hu

Transcription

1 Review of Math 727, Probability c Fall 203 by Professor Yaozhog Hu Some Cocepts: Sample space, evet, uio, itersectio, complemet, -algebra of evets, probability, 2 A collectio of subsets of S is called sigma algebra (or Borel field), deoted by if it satisfies the followig properties: (i) ;2 (ii) If A 2, the A c 2 (iii) If A,,A, 2, the [ i= A i 2 3 Give a sample space ad a associated sigma algebra, a probability fuctio is a fuctio P with domai that satisfies (a) P (A) 0 for all A 2 (b) P (A [ A 2 [ A 3 [ )=P (A )+P (A 2 )+P (A 3 )+, where the set A i, i =, 2, 3, are pairwise disjoit, (c) P ( ) =, the P is called a Probability set fuctio of the outcome of radom experimet 4 Properties of probability set fuctio () P (C )= P (C), (2) P (;) =0, (3) If C ad C 2 are subsets of such that C C 2, the P (C ) apple P (C 2 ) (4) For each C, P (C) 2 [0, ] () P (C [ C 2 )=P(C )+P(C 2 ) P (C \ C 2 ) (6) P (A \ B) P (A)+P(B) (7) P (A) = P i= (A \ C i) for ay partitio of C,,C, of (8) P ([ i= A i) apple P i= P (A i) If a job cosists of k separate tasks, the i-th of which ca be doe i i ways, i =,,k,the the etire job ca be doe i 2 k ways 6 umber of permutatios of objects chose r P k =! ( r)! (it is also deoted by k ) umber of combiatio of objects chose r C k = r =! r!( r)! possible methods of coutig of objects chose r without replacemet with replacemet! ordered ( r)! r r +r r uordered The umber of ways i which a populatio of elemets ca be divided ito k ordered parts (partitioed ito k subpopulatios) of which the first cotais r elemets, the secod cotais r 2 elemets, etc (r + +r k = ) is! r!r 2! r k! Review-

2 7 Strilig formula! = p e lim!! p e = 8 The coditioal probability of A give B is P (A B) = P (AB) P (B) If [ i= B i = S ad B i are disjoit, the P (A) = X P (A B i )P (B i ) i= Bayes formula (Bayes rule) P (A B) =P (B A) P (A) P (B) P (A i B) = P (B A i )P (A i ) P j= P (B A j)p (A j ) 9 A radom variable is a fuctio from a sample space to the real umbers 0 The cumulative distributio fuctio (cdf) of a radom variable X, deoted by F X (x), is defied by F X (x) =P X (X apple x), 0 Normal approximatio of biomial Let a k = b( + k;2, /2) X a k! (z 2 ) (z ) p 2 z applekapple p 2 z2 Examples Example Place three distiguished balls ito three distiguished cells {abc - - } 0 {a bc - } 9 { - a bc} 2 {- abc - } {b ac - } 20 { - b ac} 3 { - - abc } 2 {c ab - } 2 { - c ab} 4 {ab c - } 3 {a - bc } 22 { a b c} {ac b - } 4 {b - ac } 23 { a c b} 6 {bc a - } {c - ab } 24 { b a c} 7 {ab - c } 6 {- ab c } 2 { b c a} 8 {ac - b } 7 {- ac b } 26 { c a b} 9 {bc - a } 8 {- bc a } 27 { c b a} Review-2

3 Example 2 r balls ca be placed ito cells i r di eret ways 3 balls ca be placed i 3 cells i 27 ways Example 3 r flags of di eret colors are to be displayed o poles i a row I how may ways this ca be doe ( + ) ( + r ) Example 4 For a umber of years the New York state lottery operated accordig to the followig scheme From the umbers, 2,, 44, a perso may pick ay six for her ticket The wiig umber is the decided by radomly selectig six umbers from the forty four Fid the probability of wiig for each chose umber /( ) Example Cosider choosig a five-card poker had from a stadard deck of 2 playig cards (i) (ii) (iii) What is the probability of four aces? what is the probability of havig four of a kid? what is the probability of exactly oe pair? (i) 48/ 2 (ii) 348/ 2 (iii) / 2 Example 6 I samplig without replacemet, the probability for ay fixed elemet of the populatio to be icluded i a radom sale of size r is ( )P r = r P r Example 7 If i a city seve accidets occur each week, the all weeks will cotai days with two or more accidets, ad o the average oly oe week out of 6 will show a uiform distributio of oe accidet a day 7!/7 7 Example 8 A elevator starts with r =7passegers ad stops at = 0 floors What is the probability that o two passegers leave at the same floor? 0 7 /0 7 = Example 9 What is the probability that r persos have all di eret birthdays? (36) r /(36 r )=( 36 )( 2 36 ) ( r 36 ) r(r ) 36 Example 0 There are 2 3 = 63, 03, 9, 600 di eret hads at a bridge ad 2 = 2, 98, 960 di eret hads at poker The probability that a had at poker cotais five di eret face values is = 007 Example Each of the 0 states has two seators Cosider a committee of 0 seators (a) a give state is represeted (b) quad all states are preseted Review-3

4 (b) (a) = Example 2 Cosider a radom distributio of r balls ito cells Fid the probability of a specified cell cotai exactly k balls ( r k )( ) r k r Example 3 The umber of shortest polygoal paths (with horizotal ad vertical segmets) joiig two 6 diagoally opposite vertices of a chessboard equals 8 = 2, 870 Example 4 (Feller): I a bridge what is the probability that each player has a ace? 4! 48! (2!) 4 2! (3!) 4 00 Example Throw 2 dice oce What is the probability that each face appears exactly twice? 2! (2!) Example 6 (quality ispectio) I idustrial quality cotrol, lots of size are subjected t samplig ispectio A sample of size r is take, ad the umber k of defective items i it is determied We eed to estimate the umber of defectives i the populatio Example 7 (Estimatio of the size of a aimal populatio from recapture data) Suppose 000 is caught i a lake are marked by a red spots ad released After a while a ew catch of 000 fish is made, ad it is foud that 00 amog them have red spots What is the coclusio ca be draw cocerig the umber of fish i the lake? Whe = 900, q,00 = Maximize this q,00, we obtai ˆ = 0000 Geeralizatio to m groups q,00 = Example 8 (bridge) What is the probability of a had cosistig of five spades, four hearts, three diamods, ad oe club? Example 9 Radomly place balls ito cells util for the first time a ball is placed to a cell already occupied Fid the probability that this eeds r placemets Review-4

5 q r = The probability that this process lasts for more tha r steps is p r = (q + + q r )= r 2 r r Example 20 Cotiuously place balls ito cells at radom as log as oe particular cell remais empty The probability that this process eeds r steps is ( ) r r The probability that this process last more tha r steps is p r = (q + + q r)=( )r The media of the distributio p r is that value of r for which p + +p r apple /2 ad p + +p r +p r+ > /2 Example 2 (i) Fid the probability of four aces i a deck of 2 cards by usig the coditioal expectatio (ii) Fid the probability that havig 4 aces give that oe has already i aces i i cards (i =, 2, 3) (i=3) (i) (ii) i / 2 i (i=), ( (i=2), Example 22 Whe coded messages are set, there are sometimes errors i trasmissio I particular, Morse code uses dots ad dashes, which are kow to occur i the proportio of 3:4Thismeasthat for ay give symbol, P (dotset) = 3 7, P(dashset) =4 7 Suppose there is iferece o the trasmissio lie, ad with probability 8, a dot is mistakely received as a dash, ad vice versa If we receive a dot, what is the probability that the sigal set is dot? 2/2 Example 23 (Polya Ur problem) A ur cotais b black balls ad r red balls A ball is draw at radom It is replaced ad, moreover, c balls of the same color ad d balls of the opposite color are added A ew radom drawig is made from the ur, ad this procedure is repeated Polya ur problem is whe c>0 ad d =0 I draws the probability distributio of of exactly black balls (ad 2 = red balls is b(b + c)(b +2c) (b +( )c)r(r + c)(r +2c) (r +( 2 )c) p, 2 = (b + r + c)(b + r +2c) (b + r +( )c) = +b/c 2 +r/c 2 +(b+r)/c b/c r/c = 2 (b+r)/c Review-

6 Example 24 (Ehrefest model of heat excahge betwee two cotaiers) There are cotaiers cotaiig k particles i total Oe particle is chose at radom ad moved to aother cotaier This procedure is repeated What is the distributio of the particles after step Thik the ur cotais k balls (k black balls correspodig to the particles i the first cotaier Example 2 Imagie a collectio of N + urs, each cotai a total of N red ad white balls; the ru k cotais k red balls ad N k white balls A ur is chose ra radom ad radom draws are made from it, the ball draw beig replaced each time Suppose that all balls tur out to be red (evet A) What is the probability that the ext draw will also yield red ball? As N become large, we have P (A) = N N (N + ) P (AB) =P (B) = N + N + (N + ) P (A) Z 0 x ds = + Example 26 What is the probability of gettig at least oe 6 i 4 rolls (/6) 4 =08 Example 27 Cosider the experimet of tossig a coi three times Let H i deote the evet that the i-th toss is head The H, H 2 ad H 3 are mutually idepedet Example 28 Let S be the sample of tossig two dice Let S = {(, ), (, 2),, (6, 6)} A = {double appear} = {(, ), (2, 2), (3, 3), (4, 4), (, ), (6, 6)}, B = {the sum is betwee 7 ad 0} C = {the sum is 2 or 7, orad0} The P (A) = 6, P(B) = 2, P(C) = 3 Furthermore P (A \ B \ C) = 36 = P (A)P (B)P (C) But P (BC) =P (sum equals to 7 or 8) = /36 6= P (B)P (C) Example 29 Let S be the sample Let 8 < S = : P (A i )=/3 P (A \ A 2 )=P (A \ A 3 )=P (A 2 \ A 3 )=/9 But P (A \ A 2 \ A 3 )=/9 =6= P (A )P (A 2 )P (A 3 ) aaa bbb ccc abc bca cba acb bac cab 9 = A i = {ith place i the triple is occupied by a} ; Review-6

7 Example 30 (i) Toss two dice Let X be the sum of the umbers (ii) (iii) Toss a coi 2 times let X be the umber of heads Apply di eret amouts of amouts of fertilizer to coi plats let X be the yield / acre Example 3 I a opiio poll of 0 people about whether they agree or disagree with a certai issue let X be the umber of people who agree with the issue Fid the distributio of X P (X = i) = 0 i /2 0, i =0,,, 0 Example 32 Toss a coi three times ad let X be the umber of heads observed The cdf is 8 0 if <x<0 >< 8 if 0 apple x< 2 if apple x<2 7 >: 8 if 2 apple x<3 if 3 apple x< Example 33 (Tossig for a head) Toss a coi (p=probability of observig head) util a head appear Let X be the umber of tosses required to get a head The P (X = x) =( p) x p geometric distributio The cdf is P (X apple x) = ( p) x Example 34 Place three balls ito three cells Let N be the umber of cells occupied ad X i be the umber of balls i the i-th cell Fid the joit pdf of (N,X ) ad the joit pdf of (X,X 2 ) N X Margial dist of N 2/ /27 /9 2 6/27 6/27 6/27 0 2/ / /9 Margial Dist of X 8/27 2/27 6/27 /27 X 2 X Margial dist of X 2 0 /27 3/27 3/27 /27 8/27 3/27 6/27 3/27 0 2/27 2 3/27 3/ /27 3 / /27 Margial Dist of X 8/27 2/27 6/27 /27 Example 3 secod momets of umber whe roll a die Example 36 If the possible values of a radom variable is ±c, each with probability /2 The variace is c 2 Example 37 The variace of a symmetric die is 3/2 Example 38 The variace of a Poisso distributio is Example 39 The variace of a Biomial distributio is pq Example 40 (card matchig) A deck of umbered cards is put ito a radom order s that all! arragemet have equal probabilities The umber of matches (card i their atural place) is a radom variable S which assumes the values 0,, 2,, We kow P (S = k) = k! + 2! 3! + 4! + ( k)! (P (S = ) = 0 ad P (S = ) =! ) Fid the mea ad variace of S Review-7

8 Let X k deote the radom variable whose value is or 0 accordig to as card umber k is or ot at the k-th place The S = X + + X E (X k )= E (X2 k )= 2 k = E (X 2 j X k )= ( ) ad cov(x j X)k) = 2 ( ) Thus 2 S = 8 Example 4 (Samplig without replacemet) Hypergeometric distributio Suppose that a populatio cosists of b black ad g gree elemets, ad that a radom sample of size r is take (without repetitio) The umber S k of black elemets S r is a radom variables give by the hypergeometric distributio Fid the mea ad variace of S r Let X k deote the radom variable whose value is if the k selectio is black ad 0 if gree The S r = X + + X r E (X k )= b b + g, var(x bg k)= (b + g) 2 Thus E (S r )= E (X j X k )= b(b ) (b + g)(b + g ) rb b + g, var(s r)= rbg (b + g) 2 r b + g Example 42 Telephoe trukig problem A telephoe exchage A is to serve 2000 subscribers i a earby exchage B Assume that durig a busy hour of the day each subscriber requires a truklie to B for a average of 2 miutes We wat to istall N lies such that oe out of 00 calls will fail to fid a idle truklie immediately at disposal What is N p =2/60 = /30 P (S 2000 N) < 00 S 2000 is the umber of busy truklies Review-8