UNIT-1 FREQUENCY DISTRIBUTION

Transcription

1 Stuctu: UNIT- FREQUENCY DISTRIBUTION. Itoductio. Objctivs. Msus of Ctl Tdc.. Aithmtic m.. Mdi.. Mod..4 Empiicl ltio mog mod, mdi d mod..5 Gomtic m..6 Hmoic m. Ptitio vlus.. Qutils.. Dcils.. Pctils.4 Msus of dispsio.4. Rg.4. Smi-itqutil g.4. M dvitio.4.4 Stdd dvitio..5 Gomtic m.5 Absolut d ltiv msu of dispsio.6 Momts.7 Kl Pso s β d γ cofficits.8 Skwss.9 Kutosis. Lt us sum up. Chck ou pogss : Th k.

2 . INTRODUCTION Accodig to Simpso d Kfk msu of ctl tdc is tpicl vlu oud which oth figus gggt. Accodig to Coto d Cowd A vg is sigl vlu withi th g of th dt tht is usd to pst ll th vlus i th sis. Sic vg is somwh withi th g of dt, it is somtims clld msu of ctl vlu.. OBJECTIVES Th mi im of this uit is to stud th fquc distibutio. Aft goig though this uit ou should b bl to : dscib msus of ctl tdc ; clcult m, mod, mdi, G.M., H.M. ; fid out ptitio vlus lik qutils, dcils, pctils tc; kow bout msus of dispsio lik g, smi-it-qutil g, m dvitio, stdd dvitio; clcult momts, Kls Psio s β d γ cofficits, skwss, kutosis.. MEASUIRES OF CENTRAL TENDENCY Th followig th fiv msus of vg o ctl tdc tht i commo us : i Aithmtic vg o ithmtic m o simpl m ii Mdi iii Mod iv Gomtic m v Hmoic m Aithmtic m, Gomtic m d Hmoic ms usull clld Mthmticl vgs whil Mod d Mdi clld Positiol vgs.

3 .. ARITHMETIC MEAN To fid th ithmtic m, dd th vlus of ll tms d thm divid sum b th umb of tms, th quotit is th ithmtic m. Th th mthods to fid th m : i Dict mthod: I idividul sis of obsvtios,, th ithmtic m is obtid b followig fomul AM.. ii Shot-cut mthod: This mthod is usd to mk th clcultios simpl. Lt A b ssumd m o ssumd umb, d th dvitio of th ithmtic m, th w hv fd M. A d=-a N iiistp dvitio mthod: If i fquc tbl th clss itvls hv qul width, s i th it is covit to us th followig fomul. fu M A i wh u=-a/ i,d i is lgth of th itvl, A is th ssumd m. Empl. Comput th ithmtic m of th followig b dict d shot -cut mthods both: Clss Fqbc Solutio. Clss Mid Vlu f f d= -A f d A = Totl N = f = 46 f d = B dict mthod B shot cut mthod. Lt ssumd m A= 45. M = f/n = 46/ = 46.

4 4 M = A + fd /N = 45+/ = 46. Empl Comput th m of th followig fquc distibutio usig stp dvitio mthod. : Clss Fquc Solutio. Clss Mid-Vlu f d=-a u = -A/i fu A=8.5 i= Totl N = fu = -58 Lt th ssumd m A= 8.5, th M = A + i fu /N = / = / = =.6 PROPERTIES OF ARITHMETIC MEAN Popt Th lgbic sum of th dvitios of ll th vits fom thi ithmtic m is zo. Poof. Lt X, X, X b th vlus of th vits d thi cospodig fqucis b f, f,, f spctivl. Lt i b th dvitio of th vit Xi fom th m M, wh i =,,,. Th X i = X i M, i =,,,. fii fi XiM i i = M fi M fi i i

5 5 = Ecis Q. Mks obtid b 9 studts i sttistics giv blow clcult th ithmtic m. Q. Clcult th ithmtic m of th followig distibutio Vit : Fquc: Q. Fid th m of th followig distibutio Vit : Fquc: MEDIAN Th mdi is dfid s th msu of th ctl tm, wh th giv tms i.., vlus of th vit gd i th scdig o dscdig od of mgituds. I oth wods th mdi is vlu of th vit fo which totl of th fqucis bov this vlu is qul to th totl of th fqucis blow this vlu. Du to Co, Th mdi is th vlu of th vibl which divids th goup ito two qul pts o pt compisig ll vlus gt, d th oth ll vlus lss th th mdi. Fo mpl. Th mks obtid, b sv studts i pp of Sttistics 5,,,, 4, 9, 48 th mimum mks big 5, th th mdi is sic it is th vlu of th 4 th tm, which is situtd such tht th mks of st, d d d studts lss th this vlu d thos of 5 th, 6 th d 7 th studts gt th this vlu. COMPUTATION OF MEDIAN Mdi i idividul sis. Lt b th umb of vlus of vit i.. totl of ll fqucis. Fist of ll w wit th vlus of th vit i.., th tms i scdig o dscdig od of mgituds H two css is:

6 6 Cs. If is odd th vlu of +/ th tm givs th mdi. Cs. If is v th th two ctl tms i.., / th d ths two vlus givs th mdi. th Th m of b Mdi i cotiuous sis o goupd sis. I this cs, th mdi M d is computd b th followig fomul cf M d l i f Wh M d = mdi l = low limit of mdi clss cf = totl of ll fqucis bfo mdi clss f = fquc of mdi clss i = clss width of mdi clss. Empl Accodig to th csus of 99, followig th popultio figu, i thousds, of citis : 4, 5, 67, 8, 7, 65, 57, 488,, 7. Fid th mdi. Solutio. Agig th tms i scdig od. 488, 57, 65, 7, 5, 4, 67, 8,. H =, thfo th mdi is th m of th msu of th 5 th d 6 th tms. H 5 th tm is 5 d 6 th tm is 4. Mdi Md = 5+4/ Thousds = 5 Thousds Empls. Fid th mdi fo th followig distibutio: Wgs i Rs No. of woks Solutio. W shll clcult th cumultiv fqucis.

7 7 Wgs i Rs. No. of Woks f Cumultiv Fqucis c.f H N = 6. Thfo mdi is th msu of N + / th tm i. 8 st tm. Cll 8 st tm is situtd i th clss -. Thus - is th mdi clss. Cosqutl. cf Mdi M d l i f = + ½ 6 6 / 46 = + 5/46 = = Empl. Fid th mdi of th followig fquc distibutio: Mks No. of studts Mks No. of studts Lss th 5 Lss th 5 6 Lss th 5 Lss th 6 Lss th 6 Lss th 7 5 Lss th 4 84 Solutio. Th cumultiv fquc distibutio tbl : Clss Mks Fquc f No. of studts Cumultiv Fquc C. F Totl N = 5 Mdi = msu of 5 th tm

8 8 = 6d tm. Cll 6d tm is situtd i th clss -4. Thus mdi clss = - 4 cf Mdi M d l i f = + 5/ 6 / 4 = + 5/4 = +.4 =.4.. MODE Th wod mod is fomd fom th Fch wod L mod which ms i fshio. Accodig to D. A. L. Bowl th vlu of th gdd qutit i sttisticl goup t which th umbs gistd most umous, is clld th mod o th positio of gtst dsit o th pdomit vlu. Mod Accodig to oth sttisticis, Th vlu of th vibl which occus most fqutl i th distibutio is clld th mod. Th mod of distibutio is th vlu oud th itms tds to b most hvil cocttd. It m b gdd t th most tpicl vlu of th sis. Dfiitio. Th mod is tht vlu o siz of th vit fo which th fquc is mimum o th poit of mimum fquc o th poit of mimum dsit. I oth wods, th mod is th mimum odit of th idl cuv which givs th closst fit to th ctul distibutio.

9 9 Mthod to Comput th mod: Wh th vlus o msus of ll th tms o itms giv. I this cs th mod is th vlu o siz of th tm o itm which occus most fqutl. Empl. Fid th mod fom th followig siz of shos Siz of shos Fquc H mimum fquc is whos tm vlu is 6. Hc th mod is modl siz umb 6. b I cotiuous fquc distibutio th computtio of mod is do b th followig fomul f f Mod M l i f f f i l = low limit of clss, f = fquc of modl clss, f =fquc of th clss just pcdig to th modl clss, f =fquc of th clss just followig of th modl clss, i =clss itvl Empl.Comput th mod of th followig distibutio: Clss : Fquc : Solutio. H mimum fquc 7 lis i th clss-itvl -8. Thfo -8 is th modl clss. l =, f = 7, f = 6, f = 5, i = 7 f f Mod M l i f f f = = + 57 / 87 = + 4. = 5..

10 c Mthod of dtmiig mod b th mthod of goupig fqucis. This mthod is usull pplid i th css wh th two mimum fqucis gist two difft siz of itms. This mthod is lso pplid i th css wh it is possibl tht th ffct of ighboig fqucis o th siz of itm of mimum fquc m b gt. Th mthod is s follows : Fistl th itms gd i scdig o dscdig od d cospodig fqucis witt gist thm.th fqucis th goupd i two d th i ths d th is fous if css. I th fist stg of goupig, th goupd i.., fqucis ddd b tkig, fist d scod, thid d fouth,,. Aft it, th fqucis ddd i ths. Th fqucis ddd i th followig two ws :. i Fist d scod, thid d fouth, fifth d sith, svth d ighth, ii Scod d thid, fouth d fifth,. i Fist, scod d thid; fouth, fifth d sith, ii Scod, thid d fouth; fifth, sith d svth, iii Thid, fouth d fifth; sith svth d ighth, Now th itms with mimum fqucis slctd d th itm which cotis th mimum is clld th mod. Fo illusttio s followig mpl. Empl. Comput th mod fom th followig distibutio : Siz of Itm Fquc Solutio. Fom th giv dt w obsv tht siz hs th mimum fquc 5, but it is possibl tht th ffct of ighboig fqucis o th siz of th itm m b gt. Thus it m hpp tht th fqucis of siz o m b gt d m ot mi mod. W shll ppl th mthod of goupig. Siz of Itms I II III IV V VI

11 W hv usd bckts gist th fqucis which hv b goupd. Now w shll fid th siz of th itm cotiig mimum fquc : Colum Siz of itm hvig mimum fquc I II, III 9, IV,, V 8,9, VI 9,, H siz 8 occus tim, 9 occus tims, occus 5 tims, occus 4 tims, occus tim. Sic occus mimum umb of tims 5 tims. Hc th quid mod is siz...4 EMPIRICAL RELATION BETWEEN MEDAIN AND MODE Fo modtl smmticl distibutio o fo smmticl cuv, th ltio M Mod = M - Mdi, ppoimtl holds. I such cs, fist vlut m d mdi d th mod is dtmid b Mod = Mdi M. If i th smmticl cuv th o th lft of mod is gt th o th ight th M < mdi < mod, i.., M < Md < M

12 Mod Mdi M Mod Mdi M M < Md < M M < Md < M If i th smmticl cuv, th o th lft of mod is lss th th o th ight th i this cs Mod < mdi, m, i.. M < Md < M. Ecis c Q. Fid th Mod of th followig modl siz umb of shos. Modl siz o. of shos :,4,,,7,6,6,7,5,6,8,9,5. Q. Comput th Mod of th followig distibutio. Clss : Fquc : GEOMETRIC MEAN If,,,. vlus of th vit, o of which is zo. Th thi gomtic m G is dfid b / G =,, If f, f,, f th fqucis of,,, spctivl, th gomtic m G is giv b G = { f f f } /N N = f + f +. + f Tkig log of, w gt Log G = /N [f log + f log + + f log ] Log G = filog i N Empl. Log G = i f log. N Comput th gomtic m of th followig distibutio:

13 Mks No. of studts Solutio. H Clss Mid-vlu Fquc Log Log Log G = log /N =.79/ =.67 G = ti-log.67 =.58 mks...6 HARMONIC MEAN Poduct F log N = f = flog =.79 Th Hmoic m of sis of vlus is th cipocl of th ithmtic ms of thi cipocls. Thus if,,, o of thm big zo is sis d H is its hmoic m th [... ] H N If f, f,, f b th fqucis of,,, o of thm big zo th hmoic m H is giv b f HM.. f Empl. Fid th hmoic m of th mks obtid i clss tst, giv blow Mks : 4 5 No. of studts: Solutio. Mks X 4 5 Fquc f / f /

14 4 Rquid hmoic m is giv b f HM.. f = 5 /.964 = 5/.964 = 5/964 =.746 mks. N = f = 5 f/ =.964 Popt. Fo two obsvtios d, w hv AH = G Wh A = ithmtic m, H = hmoic m d G = gomtic m... PARTITION VALUES If th vlus of th vit gd i scdig o dscdig od of mgituds th w hv s bov tht mdi is tht vlu of th vit which divids th totl fqucis i two qul pts. Simill th giv sis c b dividd ito fou, t d hudd qul ps. Th vlus of th vit dividig ito fou qul pts clld Qutil, ito t qul pts clld Dcil d ito hudd qul pts clld Pctil... QUARTILES : Dfiitio. Th vlus of th vit which divid th totl fquc ito fou qul pts, clld qutils. Tht vlu of th vit which divids th totl fquc ito two qul pts is clld mdi. Th low qutil o fist qutil dotd b Q divids th fquc btw th lowst vlu d th mdi ito two qul pts d simill th upp qutil o thid qutil dotd b Q divids th fquc btw th mdi d th gtst vlu ito two qul pts. Th fomuls fo computtio of qutils giv b cf cf Q 4 l i, Q 4 l i f f wh l, cf,, f, i. hv th sm mig s i th fomul fo mdi... DECILES :

15 5 Dfiitio,. Th vlus of th vit which divid th totl fquc ito t qul pts clld dcils. Th fomuls fo computtio giv b cf cf D l i, D l i tc f f.. PERCENTILES : Dfiitio. Th vlus of th vit which divid th totl fquc ito hudd qul pts, t clld pctils. Th fomuls fo computtio : 7 cf cf P l i, P 7 l i tc... f f Empl. Comput th low d upp qutils, fouth dcil d 7 th pctil fo th followig distibutio: Mks goup No. of studts Mks goup No. of studts Solutio. Fist w mk th cumultiv fquc tbl : Clss Fquc Cumultiv Fquc Clss Fquc Cumultiv Fquc i To comput Q. H N = 49, ¼ N = ¼ 9 =.5 which cll lis i 5- Thus 5- is low qutil clss. l = 5, cf =, f = 5, i = -5 = 5 cf Q 4 l i f = / 5 5 = = ii To Comput Q. H ¾ N = ¾ 49 = 6.75 which cll lis i th clss 5-. Thus l = 5,cf = 6, f = 5, i = -5 = 5

16 6 cf Q 4 l i f = / 5 5 = = 5.75 iiito comput D 4 H 4/ N = 4/ 49 = 9.6, which cll lis i th clss 5-. Thus l = 5, cf =, f = 5, i = 5 4 cf D 4 l i f = / 5 5 = = 7.87 iv To comput P 7. H 7N/ = 7/ 49 = 4. which cll lis i th clss - 5. Thus l =, cf = 6, f =, i = 5 7 cf P 7 l i f = / 5 = = 4.5 Ecis b Q. Fid th mdi of th followig Q. Clcult th mdi, low d upp qutils, thid dcil d 6 th pctil fo th followig distibutio. Clss : Fquc : MEASURES OF DISPERSION DISPERSION OR VARIATION A vgs givs id of ctl tdc of th giv distibutio but it is css to kow how th vits clustd oud o scttd w fom th vg. To pli it mo cll cosid th woks of two tpists who tpd th followig umb of pgs i 6 wokig ds of wk : Mo. Tus. Wd. Thus. Fi. St. Totl pgs tpist :

17 7 tpist : W s tht ch of th tpist d tpd 5 pgs i 6 wokig ds d so th vg i both th css is 5. Thus th is o diffc i th vg, but w kow tht i th fist cs th umb of pgs vis fom 5 to 5 whil i th scod cs th umb of pgs vis fom to 4. This dots tht th gtst dvitio fom th m i th fist cs is d i th scod cs it is 5 i.., th is diffc btw th two sis., Th vitio of this tp is tmd sctt o dispsio o spd. Dfiitio. Th dg to which umicl dt td to spd bout vg vlu is clld vitio o dispsio o spd of th dt. Vious msus of dispsio o vitio vilbl, th most commo th followig..4. THE RANGE It is th simplst possibl msu of dispsio. Th g of st of umbs dt is th diffc btw th lgst d th lst umbs i th st i.. vlus of th vibl. If this diffc is smll th th sis of umbs is supposd gul d if this diffc is lg th th sis is supposd to b igul. Empl : Comput th g fo th followig obsvtio Solutio: Rg = Lgst Smllst i.., 5-5=.4. SEMI-INTER-QUARTILE RANGE Dfiitio. Th it qutil g of st of dt is dfid b It-qutil g = Q -Q wh Q d Q spctivl th fist d thid qutils fo th dt. Smi-it qutil g o qutil dvitio is dotd b Q d is dfid b Q =Q Q/ wh Q d Q hv th sm mig s giv bov. Th smi-it-qutil g is btt msu of dispsio th th g d is sil computd. Its dwbck is tht it dos ot tk ito ccout ll th itms.

18 8.4. MEAN DEVIATION Dfiitio. Th vg o m dvitio bout poit M, of st of N umbs,,, N is dfid b M Dvitio M. D. = δ m = i N i M wh M is th m o mdi o mod ccodig s th m dvitio fom th m o mdi o mod is to b computd, l i M l psts th bsolut o umicl vlu. Thus l-5l = 5. If,,, k occu with fqucis f,f,,f k spctivl, th th m dvitio δ m is dfid b δ m = k fj j M f M N N j M dvitio dpds o ll th vlus of th vibls d thfo it is btt msu of dispsio th th g o th qutil dvitio. Sic sigs of th dvitios igod bcus ll dvitios tk positiv, som tificilit is ctd. I cs of goupd fquc distibutio th mid-vlus tk s. Empl. distibutio : Fid th m dvitio fom th ithmtic m of th followig Mks : No. of studts : Solutio. Lt ssumd m A = 5 d i= Clss Mid vlu Fquc A fu -M f l-ml X f u i Totl f =5 fu = f l M l = 47

19 9 Aithmtic m M = fu u i = 5 + /5 = 7. N Th quid m dvitio fom ithmtic m δ m = f M = 47 / 5 = 9.44 N Empl. Comput th smi-it-qutil g of th mks of 6 studts i Mthmtics giv blow : Mks Goup No. of Studts Mks Goup No. of studts Solutio. Mks Goup Fquc f Cumultiv Fquc c.f f = 6 To clcult low Qutil Q. H N = 6. So ¼ N+th i.., 6 th studts lis i th mks goup -. Thus low qutil clss is -. Q = 4 N F l i = f 5.75 =.75.

20 Simill Q = = Smi-it qutil g = Q Q = ½ =. Mks.4.4 STANDARD DEVIATION Root m squ Dvitio : Dfiitio. It is dfid s th positiv squ oot of th m of th squs of th dvitios fom oigi A d dotd b s s f A N { } M squ dvitio. It is dfid s th m of th squs of th dvitios fom oigi A. Thus s = f A Rmk. Th oigi A m b tk t bit poit A. Stdd Dvitio : Dfiitio. Stdd dvitio o S.D. is th positiv squ oot of th ithmtic m of th squ dvitios of vious vlus fom thi ithmtic m M. It is usull dotd b σ. Thus σ = s f M N { } Rmks:. Wh th dvitios is clcultd fom th ithmtic m M, th oot m squ dvitio bcoms stdd dvitio.

21 . Th squ of th stdd dvitio σ is clld vic.. Th qutit s is sid to b scod momt bout th vlu A d is dotd b µ. 4. Th vic σ is clld th scod momt bout th m M d is dotd b µ. RELATION BETWEEN STANDARD DEVIATION AND ROOT-MEAN SQUARE DEVIATION Cosid fquc distibutio disct sis : f: f f f Lt A b th ssumd m d M th ithmtic m. Also suppos M-A = d. Th f M f A M A N N σ N f d whx A, d M A N N N f d. fx d. f f A d. f A d N N f A d f fa d N N N f A d M A d N f A d s d N Hc s = σ + d Rltio shows tht s is lst wh d = i.., A = M d th lst vlu of s is qul to σ. I oth wods th stdd dvitio is th lst possibl oot m squ dvitio. Rmk. Sic d >, lws, thfo, fom, w hv S > σ i.., m squ dvitio bout poit A is gt th vic.

22 SHORT CUT METHOD FOR CALCULATING STANDARD DEVIATION W kow tht σ f { f } N N Wh ξ = -A d A = ssumd m STEP DEVIATION METHOD TO COMPUTE S.D. If u = -A/h = ξ/h, th ξ = uh. B shot cut mthod σ fu h { fu } N N σ = h N fu N [{ f u }].5 ABSOLUTE AND RELATIVE MEASURES OF DISPERSION Th msu of dispsios ml g, qutil dvitios, it-qutil dvitio. M dvitio, stdd dvitio, oot m squ dvitio ths hv b discussd bov sid to b bsolut msu of dispsio, sic th pssd i tms of uits of obsvtios Cm., Km., Rs., dg tc.. W kow tht difft uits c ot b compd; fo mpl ctimt c ot b compd with up. Thfo, th dispsios i difft uits c ot b compd. Also th msus of dispsio dpd o th msus of ctl tdc. Thfo, it is dd to dfi som msus which idpdt of th uits of msumt d c b djustd fo msus of ctl tdc. Such tp of msus clld ltio msus of dispsio o cofficits of dispsio. Ths ltiv msus pu umbs d usull pssd s pctgs. Th usful to comp two sis i difft uits d lso to comp vitios of two sis hvig difft mgituds. Som of th ltiv msus of dispsio o cofficit of dispsio which i commo us giv blow : i Qutil cofficit of dispsio. It is usull dotd b Q.D. d is dfid b

23 Q Q.D.= Q Q Q ii Cofficit of m dispsio = M dvitio bout poit /. H poit c b plcd b m, mdi, mod tc. iii Cofficit of vitio o cofficit of dispsio. It is dfid b th tio σ/m, wh σ is stdd dvitio d M is th ithmtic m. It is dotd b C.V. o V. thus C.V. o V = σ / M Somtims, w dfi C.V. o V = σ / M Empl. Clcult th S.D. d cofficit of vitio C.V. fo th followig tbl : Clss : Fquc : Solutio. W pp th followig tbl fo th computtio of S.D. Clss Mid-vlu f 5 fu fu u N= f = 4 fu = 65 fu = 85 Lt ssumd m = 5 = A s d h = A.M., M = A + h fu/n = /4 = = 9.64 S.D., σ = h fu fu [ ] N N

24 4 85 = [.464 ] 4 = [.75.5] =.55 =.59 = 5.9 C.V. = σ/m = 5.9/9.64 = 4.%..6 MOMENTS Fo fquc distibutio, th th momt bout poit A is dfid s th ithmtic m of th pows of dvitios fom th poit A..7. Momts bout m o Ctl Momts : i Fo idividul sis. Lt,,, b th vlus of th vibl, th th th momt bout th m ithmtic m is dotd b µ d is dfid b ii i µ = i, fo =,,,,. Fo fquc distibutio. Lt : f: f f f b disct fquc distibutio. Th th th momt µ bout th m is dfid b i µ = i N i µ = N o Pticul Css. Fo =,, fo =,,,,. Wh i fi N, fo =,,,,. Wh fi N fi i fi i i N µ = N N N Hc fo ll distibutios,

25 5 µ = Fo =, µ = fi i N fii fi. N N N N Hc fo ll distibutios, µ = Fo =, µ = f i i = σ = vic. N Hc fo ll distibutios, w hv µ = stdd dvitio = Vic Fo =, µ = f i, d so o. i N MOMENTS ABOUT ANY POINT RAW MOMENTS : Fo fquc distibutio th th momt bout poit = A, is dfid s th ithmtic m of th th pows of th dvitios fom th poit =A d is dotd b µ. If X : F: f f f B disct fquc distibutio, th µ = N i fi i A, =,,,,, d i fi N. I cs of idividul sis i A µ = i,,,,,...

26 6 Pticul css. Fo =, = µ N f A f N N N i i i i i Fo = = µ f A f A f N N N i i i i i i i i = A/N N = A = d s Wh = Aithmtic m of giv dt. Fo = = µ fi i A fi{ i A} i N i N A f N N i i i i i f Fo = = σ + -A = σ + d. = µ N fi i, d so o. i : RELATION BETWEEN CENTRAL MOMENTS µ AND MOMENTS ABOUT ANY POINT µ W hv: µ C C... MOMENTS ABOUT THE ORIGIN : If : f: f f f

27 7 b disct fquc distibutio, th th th momt bout th oigi is dotd b V, s d is dfid b V = N i f i i, =,,,, d f i = N i Puttig =,,,, w gt V f i = N N = N i V N i f i i N i f i { i + } f i { i - + C i C i } N i V = µ C µ C µ.7 KARL PEARSON S β AND γ COEFFICIENTS Kl Pso gv th followig fou cofficits. Clcultd fom th ctl momts, which dfid s Bt coficits β = µ /µ β = µ 4 /µ Gmm cofficits γ = γ = β = 4 Th sig of γ dpds upo µ is positiv th γ is positiv. If µ is gtiv th γ is gtiv. Th bov fou cofficits pu umbs d thus do ot hv uit. Th β d γ cofficits giv som id bout th shp of th cuv obtid fom th fquc distibutio. This w shll discuss i th topic Kutosis d Skw ss. Empl. Clcult th fist fou ctl momts fom th followig dt Clss : Fquc : 5 7 4

28 8 Solutio. Lt A=5. H h=. To fcilitt th clcultios, lt u = A 5 h Clss Fquc f N = f = Midvlu U= 5 fu fu fu fu fu = fu = fu = 8 fu 4 = 9 Th th momt bout poit =A=5 is giv b µ = µ = µ = µ = µ 4 = h h h h h fu N fu N fu N fu N fu N 4 4 = / = 5 = / = 5 = 8/ = 4 = 9/ = 45. To Clcult Ctl Momts. µ = lws µ = µ µ = 5 5 = 5 µ = µ - µ µ + µ = = -6 µ 4 = µ 4-4 µ µ + 6 µ µ µ 4 = = 765

29 9.8. SKEWNESS B skwss i som fquc distibutio, w m th lck i smmt. [If th fqucis smmticll distibutd bout th m, th th distibutio is clld smmticl, i oth wods, distibutio is clld smmticl wh th vlus quidistt fom th m hv qul fqucis.] Skwss is lso tmd s smmt. Skwss dots th tdc of distibutio to dpt fom smmt. Accodig to Simpso, Skwss o smmt is th ttibut of fquc distibutio tht tds futh o o sid of th clss with th highst fquc tht o th oth. M=M =M d W kow tht fo smmticl distibutio th m, mdi d mod coicid. Thfo, skwss i distibutio is show wh ths th vgs do ot coicid. Skwss idicts tht th fquc cuv hs log til o o sid fo th vg. Wh th fquc cuv hs log til o ight sid, th skwss is clld positiv. Wh th fquc cuv hs log til o lft sid, th skwss is clld gtiv. I oth wods, th skwss is positiv if M<Md<M d gtiv if M<Md<M, wh M, Md d M m, mdi d mod spctivl. MEASURE OF SKEWNESS : W shll giv followig th msus to msu th skwss : i Fist cofficit of skwss. It is lso kow s Bowl s cofficit of skwss d is dfid s Q Q M d Q M d M d Q Cofficit of skwss = JQ. Q Q Q M M Q d d Wh Q d Q low d upp qutils spctivl d Md is mdi. Cll this msu is bsd o th fct tht i skw cuv, th mdi dos ot li hlf w btw Q d Q. This fomul fo cofficit of skwss is usd wh mod is wll dfid

30 Scod cofficit of skwss. It is lso clld Kl Pso s cofficit of skwss d is dfid s M Mod M M Cofficit of skwss= = J S t dddvitio If mod is ot wll dfid, th Coff. Of skwss = M M d Cll this msu is bsd o th fct tht m d mod ot coicidt. Not tht both of th bov cofficits pu umbs sic both th umto d domito hv th sm dimsios. ii Cofficits of Skwss Bsd o Momts. Wh th is smmticl distibutio, ll th momts of odd od bout th ithmtic m i.., µ,µ,µ5 tc. vish. If th vlus of ths cofficits do ot vish th th is skwss i th fquc distibutio. Accodig to Kls Pso th cofficits of skwss ctl giv b th followig fomul : Fist Cofficit of Skwss = Scod Cofficit of Skwss = If skwss i th sis is v smll th scod cofficit of skwss should b usd. Cofficits of skwss bsd o momts lso clld Momt Cofficit of skwss. Empls. Comput th Bowl s cofficit of skwss fo th followig fquc distibutio : Mks : No. of studts : 7 5 Solutio. Mks Fquc C.F

31 Low qutil, Q = N/4 th tm = 7/4 th tm = 6.75 th tm. Q = 4 N F 6.75 l i = f Q = = = 6.79 mks.mdi, 7 Md = N/th tm =.5 th tm Md = 4 N F.5 9 l I = = = 4.5 mks. f Q = N/4th tm =.5 th tm. Q = 4 N F l i f.5 9 Q = = +.5 =.5 mks. 5 Bowl s cofficit of skwss QQ M JQ = d Q Q = KURTOSIS I Gk lgug kutosis ms bulgis.. kutosis idicts th tu of th vt of th cuv. Svl sttisticis dfid kutosis. Som of ths dfiitios : I sttistics, kutosis fs to th dg of fltss of pkd ss i th gio bout th mod of fquc cuv. Th dg of kutosis of distibutio is msud ltiv to th pkd ss of oml cuv. A msu of kutosis idicts th dg to which cuv of th fquc distibutio is pkd o flt-toppd. Kl Pso i 95 dfid followig th tps of cuvs :. Noml Cuv o Msokutic Cuv. A cuv which is ith flt o pkd is clld oml cuv o mso-kutic cuv. Fo such tp of cuv w hv β = d γ=.

32 . Lptokutic Cuv. A cuv which is mo pkd th th oml cuv is clld lptokutic cuv. Fo such tp of cuv, w hv β> d γ>.. Pltkutic Cuv. A cuv which is mo fltt th th oml cuv is clld pltkutic cuv. Fo such tp of cuv, w hv β< d γ<. Msu of Kutosis Scod d fouth momts usd to msu kutosis. Kl Pso gv th followig fomul to msu kutosis : Kutosis o µ / µ. 4 To msu kutosis, γ is usd d it is giv b th followig fomul: γ = β = 4 Dductios. If γ =, th cuv is oml. If γ >, th cuv is lpokutic. If γ <, th cuv is pltkutic. Empl. Th fouth momt bout m of fquc distibutio is 768. Wht must b vlu of its stdd dvitio i od tht th distibutio b i Lptokutic ii Msokutic iiipltkutic. Giv µ 4 = 768. Kutosis = = µ 4 / µ = µ 4 / σ 4 = 768 / σ 4 Now th distibutio will b. Lptokutic if > => 768/ σ 4 > => σ 4 <768/ => σ 4 < 56 => σ 4 < 4 4 => σ < 4. Msokutic if = => 768 / σ 4 = => σ = 4

33 . Pltkutic if < => 768/ σ 4 < => σ >4. Lt us sum up Aft goig though this uit, ou would chivd th objctivs sttd li i th uit. Lt us cll wht w hv discussd so f * A vg is sigl vlu withi th g of th dt tht is usd to pst ll th vlus i th sis. * To fid th ithmtic m, dd th vlus of ll tms d thm divid sum b th umb of tms, th quotit is th ithmtic m. * Th mdi is th vlu of th vibl which divids th goup ito two qul pts o pt compisig ll vlus gt, d th oth ll vlus lss th th mdi * Th mod is tht vlu o siz of th vit fo which th fquc is mimum o th poit of mimum fquc o th poit of mimum dsit. I oth wods, th mod is th mimum odit of th idl cuv which givs th closst fit to th ctul distibutio. * Fo modtl smmticl distibutio o fo smmticl cuv, th ltio M Mod = M - Mdi, ppoimtl holds. I such cs, fist vlut m d mdi d th mod is dtmid b Mod = Mdi M. If i th smmticl cuv th o th lft of mod is gt th o th ight th M < mdi < mod, i.., M < Md < M * If f, f,, f th fqucis of,,, spctivl, th gomtic m G is giv b G = { f f f } /N N = f + f +. + f

34 4 * If f, f,, f b th fqucis of,,, o of thm big zo th hmoic m H is giv b HM.. f f * Th vlus of th vit which divid th totl fquc ito fou qul pts, clld qutils * Th vlus of th vit which divid th totl fquc ito t qul pts clld dcils * Th vlus of th vit which divid th totl fquc ito hudd qul pts, t clld pctils. * Th g of st of umbs dt is th diffc btw th lgst d th lst umbs i th st * Smi-it qutil g o qutil dvitio is dotd b Q d is dfid b Q =Q Q/ * If,,, k occu with fqucis f,f,,f k spctivl, th th m dvitio δ m is dfid b δ m = k fj j M f M N N j * Stdd dvitio o S.D. is th positiv squ oot of th ithmtic m of th squ dvitios of vious vlus fom thi ithmtic m M. It is usull dotd b σ.thus σ = s f M N { } * Fo fquc distibutio, th th momt bout poit A is dfid s th ithmtic m of th pows of dvitios fom th poit A. * Skwss dots th tdc of distibutio to dpt fom smmt.

35 5 * msu of kutosis idicts th dg to which cuv of th fquc distibutio is pkd o flt-toppd.. Chck ou pogss : Th k Ecis Q.Clcutt th msu of Kutosis fo th followig distibutio. Mks : No. of cdidt : Q.Th fist fou momts of th distibutio bout th vlu 5 of vibl,,4 d 5. Fid th ctl momt. Uit- Pobbilit Stuctu:. Itoductio. Objctivs. Rdom pimt. Smpl spc

36 6.4 Evt.5 Additiol lw of pobbilit.6 Coditiol pobbilit.7 Multiplictiv lw of pobbilit.8 Idpdt d dpdt vt.9 B s thom. Rdom vibl. Pobbilit dsit fuctio. Cotiuous pobbilit distibutio. Cummultiv distibutio fuctio.4 M, Mdi, Mod d Momts fo cotiuous distibutio fuctio.5 Mthmticl pcttio.6 Covic.7 Lt us sum up.8 Chck ou pogss : Th k

37 7. INTRODUCTION If coi is tossd ptdl ud sstill homogous d simil coditios, th o is ot su if hd o til will b obtid. Such tps of phom i.. phom which do ot ld itslf to dtmiistic ppoch clld updictbl o pobbilistic phom. I 99 A. N. Komogov, Russi mthmtici, tid succssfull to lt th tho of pobbilit with th st tho b iomtic ppoch. Th iomtic dfiitio of pobbilit icluds both th clssicl d th sttisticl dfiitios s pticul css d ovcoms th dfiitios of ch of thm.. OBJECTIVES Th mi im of this uit is to stud th pobbilit. Aft goig though this uit ou should b bl to : dscib dom pimts, smpl spc, dditiv lw d multiplictiv lw of pobbilit, dpdt d idpdt vts tc ; clcult m, mod, mdi, d momts fo cotiuous distibutio fuctio, mthmticl pcttio; fid out covic; kow B s thom, dom vibls, pobbilit dsit fuctio, cotiuous pobbilit distibutio, cumultiv distibutiv fuctio,tc.

38 8. RANDOM EXPERIMENT Cosid bg cotiig 4 whit d 5 blck blls. Suppos blls dw t dom. H th tul phomo is tht both blls m b whit o o whit d o blck o both blck. Thus th is pobbilistic situtio. W fl ituitivl i th followig sttmts. i Th pobbilit of gttig til i o toss of ubisd coi is ½. ii Th pobbilit of gttig c i sigl of ubisd di is /6. Simill th pobbilit of gttig o i sigl thow of ubisd di should b th sum of pobbilitis of gttig o i.. /6 + /6 = /. I oth wods, th should b th sum of pobbilistic situtios, w d mthmticl modls. A pobbilistic situtio is clld dom pimt d is dotd b E. Ech pfomc i dom pimt is clld til d th sult of til is clld outcom o smpl poit o lmt vt.. SAMPLE SPACE.

39 9 A smpl spc of dom pimt is th st of ll possibl outcoms of tht pimt d is dotd b S Fiit smpl spc. A smpl spc cotiig fiit umb of smpl poits, is clld fiit smpl spc..4 EVENT. Of ll th possibl outcoms i th smpl spc of pimt som outcoms stisf spcifid dsciptio, it is clld vt.i oth wods v o-mpt subst of smpl spc is clld vt of th smpl spc. It is dotd b E. Svl vts dotd b E, E, tc. Cti d impossibl vts. If S is smpl spc, th S d both substs of S d so S d both vts. S is clld cti vt d is clld impossibl vt. Equll likl vts. Two vts cosidd qull likl if o of thm cot b pctd i pfc to th oth. Fo mpl, if ubisd coi is tossd th w m gt of hd H o til T, thus th two difft vts qull likl.

40 4 Ehustiv Evts. All possibl outcoms i til, clld hustiv vts. Fo mpl, if ubisd di is olld, th w m obti o of th si umbs,,,4,5 d 6. Hc th si hustiv vts i this til. Fvobl vts. Th totl umb of fvobl outcoms o ws i til, to hpp vt, clld fvobl vts. Fo mpl, If pi of fi dic is tossd th th fvobl vts to gt th sum 7 si :,6,,5,,4, 4,, 5,, 6,,. Mutull Ehustiv o Icomptibl Evts. Two o mo th two vts clld mutull clusiv vts if th is o lmt o outcom o sult commo to ths vts. I oth wods, vts clld mutull clusiv if th hppig of o of thm pvts o pcluds th hppig of th oth vts. If E, E two mutull clusiv vts th E E E d E mutull clusiv. EXAMPLES ON SAMPLE SPACE AND EVENT

41 4 Empl. I sigl toss of fi di, fid smpl spc b vt of gttig v umb c vt of gttig odd umb d vt of gttig umbs gt th, vt of gttig umbs lss th 4. Solutio.. Wh w toss di, th w m gt of th si umbs,,,,4,5d 6. hc th st of ths si umbs is th smpl spc S fo this pimt, i.., S={,,,4,5,6}; I th bov pimt, to gt v umb is vt, s E ; to gt odd umb is vt, s E ; to gt umbs gt th is vt, s E ; d to gt umbs lss th 4 is vt, s E 4. Thus b E ={,4,6} c E ={,,5} d E ={4,5,6} E 4 ={,,} Empl. Cosid pimt i which two cois tossd togth. Fid th smpl spc. Fid lso th followig vts: Hds o th upp fcs of cois, bhd o o d til o oth, ctils o both,d t lst o hd. Solutio. If H dots hd d T dots til th th toss of two cois c ld to fou css H,H,T,T,H,T,T,H ll qull likl. Hc th smpl spc S is th st of ll ths fou odd pis, thus S={H,H,T,T,H,T,T,H};

42 4 I this pimt lt E, E, E d E4 b th vts of gttig both hds, o hd d o til, both tils d t lst o hd spctivl, th E ={H,H} b E ={H,T,T,H} c E ={T,T} d E 4 ={H,H,H,T,T,H} Simpl Ad Compoud Evts Cosid dom pimt d lt,,..., b th outcoms o smpl poits so tht th smpl spc S fo this pimt is giv b X={,,..., }. Lt E b vt ltd to this pimt th E S. Th st E pstig th vt, m hv ol o o mo lmts of S. Bsd upo this fct, v vt c b dividd ito followig two tps of vts: Simpl vt. If E cotis ol o lmt of th smpl spc S, th E, is clld simpl vt. Thus E i wh S, is simpl vt sic it cotis ol o lmt of S. i Compositio of vts

43 4 Followig th fudmtl uls to composit two o mo vts b th hlp of st ottios. Lt S b smpl spc d A d B b its two vts: i Th vt pstd b A B o A+B. If th vt E hpps wh A hpps o B hpps th E is dotd b A B i.., th vt E pstd b A B icluds ll thos lmts o outcoms o sults which A o B coti. A B A B ii Th vt pstd b A B o AB. If th vt E hpps wh th vts A d B both hpp th th vt E is pstd b A B i.., th vt E dotd b A B icluds lmts o outcoms commo to both A d B. Th shdd i figu psts th vt E = A B. A B.

44 44 iii Complmt of vt A o th vt A o A. If th vt E hpps wh th vt A dos ot hpp th E is dotd b A. A A.5 ADDITIVE LAW OF PROBABILITY Thom. If E d E two vts th P E E P E P E P E E. Poof. Lt S b th smpl spc d b th umb of lmts i S. Lt l b th umb of lmts i E d m th umb of lmts i th vt E, i.., S, E l, E m. if th vts E d E ot mutull clusiv th E E. Lt E E

45 45 Cll E E l m. Now th pobbilit of E o E hppig P E E E E l m l m S E E E E S S S o P E E P E P E P E E co. If E d E b mutull clusiv vts th. E E d E E Now fom, w hv P E E P E P E Empl. If 4 is th pobbilit of wiig c b th hos A d b th pobbilit of wiig th sm c b th hos B. Fid th pobbilit tht o of ths hos will wi.

46 46 Solutio. Lt E d E b th vts tht th hos A d B wis th c spctivl. Th P E, P E 4 W kow tht if th hos A wis th c th th hos B cot wi th c d if B wis th c th A c ot wi. Hc th vts E d E mutull clusiv vts. Thfo, th pobbilit tht o of A o B is th c is giv b P E E P E P E = 4 + = 7 Empl. Discuss d citiciz th followig: P A, P B, P C 6 4 fo th pobbilitis of th mutull clusiv vts A,B,C. Solutio. Sic A,B, C mutull clusiv vts, thfo: P A B C P A P B P C which is impossibl 6 4 >

47 47.6 CONDITIONAL PROBABILITY. Wh th hppig of vt E dpds upo th hppig of oth vt E th th pobbilit of E is clld coditiol pobbilit d is dotd b P E. Thus P E E E fo th vt E wh th E hs ld hppd. dots th coditiol pobbilit.7 MULTIPLICATIVE LAW OF PROBABILITY Thom. If E d E two vts, th spctiv pobbilitis of which kow, th th pobbilit tht both will hpp simultousl is th pduct of pobbilit of E d th coditiol pobbilit of E wh E hs ld occud i.., E P E E P E P. E Poof. Lt S b th smpl spc fo pimt d E d E b its two vts. Suppos th vt E hs occud d E. Sic E S d th vt E hs occud, thfo ll lmts of S cot occu d ol thos lmts of S which blog to E c occu. I this cs th ducd smpl spc will b E. Now if th vt E occus, th ll lmts of E cot occu but ol thos lmts of E which blog to E c occu. Th st of commo lmts is E E. Hc th pobbilit of E wh E hs occud E i.., P E [i.., coditiol pobbilit of E wh E hs occud] is giv b

48 48 P E E = E E E E E S E S P E E PE E P E E P E P. E Simill w c pov tht P E E P E P E E.8 INDEPENDENT AND DEPENDENT EVENTS Dfiitio. Lt E d E b two vts of smpl spc. If th occuc of E dos ot dpd o th occuc of E d th occuc of E dos ot dpd o th occuc of E o i oth wods th occuc of o dos ot dpd o th occuc of oth th E d E clld idpdt vts othwis th clld dpdt vts. Empl. if A d B two vts, wh P A, P B, d. P A B b. P B A P A B,th vlut th followig: 4

49 49 c. P A B Solutio.. P A B P B. P A/ B P A B P A B = 4 PB 4 b. P B A P A B 4 P B A P A P A 4 c. P A B P A P B P A B BAYE S THEOREM A vt B c b plid b st of hustiv d mutull clusiv hpothsis A,A,.,A. Giv pioi pobbilitis PA,PA,,PA cospodig to totl bsc of kowldg gdig th occuc of B d coditiol pobbilitis

50 5 PB/A,PB/A,,PB/A. th postio pobbilit PA / B j of som vt Aj is giv b PA / B j PA.P B / A i j PA P B / A i j i. th pobbilit PC/B of mtiliztio of oth vt C is giv b PA / B j i PA.P B / A P C / A B i i i i PA P B / A i i Poof: sic th vt B c occu wh ith A occus, o A occus, o,,a occus i., B c occu i compositio with ith A o A cosqutl B B B B B A A A... A P B P BA BA BA... BA Sic A,A A mutull clusiv, hc BA,BA, BA mutull clusiv foms, thfo b totl pobbilit thom, w hv

51 5 P B P BA P BA P BA... P BA P BA PA P B / A i i i i i Wh PB/A i is th coditiol pobbilit of B wh A i hs ld occud. Now fom th thom of compoud pobbilit, w hv PA B PA PA / B j j j PA / B Fomd w gt j PA B PA PA / B j j j P B P B PA / B j PA P B / A i i PA P B / A i i i b th futh vt C c occu i mutull clusiv css ml A C/B,,A C/B. hc th coditiol pobbilit of C is giv b P C / B PA C / B PA C / B... PA C / B = PA C / B PA / B P C / A B i i i i i

52 5 i PA P B / A P C / A B i i i i PA P B / A i i Empl. A bg cotis whit d blck blls, oth bg coti 5 whit d blck blls. If bg is slctd t dom d bll is dw fom it, fid th pobbilit tht it is whit. Solutio. Lt B b th vt of gttig o whit bll, d A,A b th vts of choosig fist bg d scod bg spctivl PA = th pobbilit of slctig th fist bg out of two bgs = simill, PA = th pobbilit of slctig scod bg = PB/A i =th coditiol pobbilit of dwig o whit bll whil fist bg hs b slctd C C = 5 5 Simill P B / A 5 / 8 PB= Th pobbilit tht whit bll is dw

53 5 = i PA P B / A i i = P P B P P B A / A A / A = As.. RANDOM VARIABLES. Dfiitio. A l vlud fuctio dfid o smpl-spc is clld dom-viblo disct dom vibl. A dom vibl c ssum ol st of l vlus d th vlus which th vibl tks dpds o th chc. Rdom vibl is lso clld stochstic vibl o simpl vit. Fo mpl. Suppos pfct di is thow th, th umb of poits o th di is dom vibl sic hs th followig two poptis: i tks ol st of disct vlus,,,4,5,6; ii th vlus which tks dpds o th chc. Actull tks vlus,,,4,5,6 ch with pobbilit /6. Th st of vlus,,,4,5,6 with thi pobbilitis /6 is clld th Pobbilit Distibutio of vit. I gl suppos tht cospodig to hustiv d mutull clusiv css obtid fom til, vit tks vlus,,..., with thi pobbilitis p, p,..., p.

54 54 Th st of vlus i fo i=, with thi pobbilitis p i fo i=, is clld th Pobbilit Distibutio of th vibl of tht til. It is to b otd tht most of th poptis of fquc distibutio will b qull pplicbl to pobbilit distibutio. Cotiuous Vit So f w hv discussd with disct vit which tks fiit st of vlus. Wh w dl with vits lik wights d tmptu th w kow tht ths vits c tk ifiit umb of vlus i giv itvl. Such tp of vits kow s cotiuous vits. Dfiitio. A vit which is ot disct, i.., which c tk ifit umb of vlus i giv itvl b, is clld cotiuous vit.. PROBABILITY DENSITY FUNCTIONS Lt X b cotiuous dom vibl d lt th pobbilit of X fllig i th ifiitsiml itvl d, d P d X d f d, b pssd b fd i.. wh f is cotiuous fuctio of X d stisfis th followig two coditios: i f

55 55 b if X b ii f d, f d, if X th th fuctio f is clld th pobbilit dsit fuctioo i bif p.d.f. of th cotiuous dom vibl X. Th cotiuous cuv Y=f is clld th Pobbilit dsit cuv o i bif pobbilit cuv. Th lgth of ifiitsiml itvl d d its mid poit is. Rmks: d, d is If th g of X b fiit, th lso it c b pssd s ifiit g. Fo mpl,.f=, fo <.f=ф, fo X b f= fo >b. Th pobbilit tht vlu of cotiuous vibl X lis withi th itvlc,d is giv b d P c X d f d c Th cotiuous vibl lws tks vlus withi giv itvl howsov smll th itvl m b. 4 If X b cotiuous dom vibl, th PX=k= wh k is costt qutit.

56 56. CONTINUOUS PROBABILITY DISTRIBUTION: Th pobbilit distibutio of cotiuous dom vit is clld th cotiuous pobbilit distibutio d it is pssd i tms of pobbilit dsit fuctio.. CUMULATIVE DISTRIBUTION FUNCTION: Th pobbilit tht th vlu of dom vit X is o lss th is clld th cumultiv distibutio fuctios of X d is usull dotd b F. I smbolic ottio, th cumultiv distibutio fuctio of disct dom vit X is giv b F P X p i i Th cumultiv distibutio fuctio of cotiuous dom vit is giv b F P X f d

57 57.4 MEAN, MEDIAN MODE AND MOMENTS FOR A CONTINUOUS DISTRIBUTION FUNCTION M M X E X = if Also m= 6 M d M d if b Gomtic M. If Gis gomtic m, th log G = log f d Elog if b if b = log f d Elog Hmoic M. Lt H b th hmoic m, th f d E H if b f d E H if b 4 Mdi. Th mdi M d is giv b Md f d f d f d Md 5 Th low qutil Q d upp qutil Q giv b Q f d = 4 d Q f d 4

58 58 Q f d = 4 d Q f d 4 6 Mod. Th mod is th vlu of th vit fo which d f d d d f d [I oth wods, th mod is th vlu of th vit fo which pobbilit f is mimum]. Th coditio fo which is tht th vlus obtid fom d/df= lis withi th giv g of. 7 Momt. Th th momt bout giv bit vlu A is giv b: m f d, if b X A f d, if b 8 Th M dvitio bout th m m. it is giv b m f d, if b m f d, if b 9 Vic. Fo cotiuous distibutio th vic is giv b m f d, if b = m f d, if b Stdd dvitios.d.. Th positiv squ oot of vic is clld S.D. d is dotd b.

59 59 Empl: Fo th distibutio df 6 d,, fid ithmtic m, hmoic m, mdi d mod. Is it smmticl distibutio? Solutio. Fo th giv distibutio w hv 6 d 6 f= 6 Aithmtic M. is p.d.f. M. f d M.6 d 6 d = Hmoic M. It is giv b.6 H d H 6 H=/ Mdi. M d is giv b: Md f d

60 6 d M d 6 M d 6 4Md 6Md Md M d Md M d o M d sic M d lis btw d. M d 4 Mod. It is tht vlu of vit fo which d / d d 6. d d d / d 6 d d d d / d 6 Sic m, mdi d mod coicid, thfo th distibutio is smmticl..5 MATHEMATICAL EXPECTATION Dfiitio. Lt b th disct dom vibl d lt its fquc distibutio b s follos : Vit :..

61 6 Pobbilit : p p p.. p wh p p p... p p, th th mthmticl pcttio of o simpl pcttio of is dotd b E d is dfid b E p p p... p p p i i i th if is pobbilit dsit fuctio cospodig to th vit, E d If is such fuctio of tht it tks vlus PA P B / A PA P B / A wh tks th vlus of,... Lt p, p... b thi spctiv pobbilitis, th th mthmticl pcttio of, dotd b E[ ], is dfid s E[ ] p p... p wh p=. Vic Th ltio E E E E dotd b V, V is [{ } ] [ ] dfid s th vic of th distibutio of. d hv th sm ltio s fo th fquc distibutio. But h th pctd vlu of th dvitio of th vit fom its m vishs i.., E[ E ].

62 6 Mthmticl Epcttio fo Cotiuous Rdom Vibl Suppos X is cotiuous dom vibl with pobbilit dsit fuctio f, th mthmticl pcttio E of with cti stictios is giv b Epcttio of Sum E f d Thom. Th pcttio of th sum of two vits is qul to th sum of thi pcttios, i.., if d two vits th E E E Poduct of Epcttios Thom. Th pcttio of th poduct of two idpdt vits is qul to th poduct of thi pcttios E E. E.6 COVARIANCE Dfiitio. Lt d b two dom vibls d d b thi pctd vlus o ms spctivl. Th co-vic btw d, dotd b cov,, is dfid s cov, E[ ]. Empl. wht is th pctd vlu of th umb dof poits tht will b obtid i sigl thow with odi di? Fid vic lso. Solutio. Th vit i.., umb showig o di ssums th vlus,,,4,5,6 d pobbilit i ch cs is /6.

63 6 giv pobbilit distibutio is s follows: : p: /6 /6 /6 /6 /6 /6 E pii 6 i = p p p... p 6 6 = [ 4 5 6] 6 6 Also v= E E[ ] [ ] 7 / 5 / 8. 6 = i Empl. Fid PA P B / A P C / A B i i i i PA P B / A distibutio:. : p : /8 /6 /8 ¼ / i i fo th followig pobbilit Solutio. Th m of pobbilit distibutio is E=.p =

64 64 E p = Th scod momt bout th oigi E E p { } =vic of th distibutio..7 LET US SUM UP Aft goig though this uit, ou would chivd th objctivs sttd li i th uit. Lt us cll wht w hv discussd so f A pobbilistic situtio is clld dom pimt. A smpl spc of dom pimt is th st of ll possibl outcoms of tht pimt d is dotd b S Of ll th possibl outcoms i th smpl spc of pimt som outcoms stisf spcifid dsciptio, it is clld vt Two vts cosidd qull likl if o of thm cot b pctd i pfc to th oth. vts clld mutull clusiv if th hppig of o of thm pvts o pcluds th hppig of th oth vts. If E d E two vts th P E E P E P E P E E.

65 65 Wh th hppig of vt E dpds upo th hppig of oth vt E th th pobbilit of E is clld coditiol pobbilit d is dotd b P E E. If E d E two vts, th spctiv pobbilitis of which kow, th th pobbilit tht both will hpp simultousl is th pduct of pobbilit of E d th coditiol pobbilit of E wh E hs ld occud i.., E P E E P E P. E If th occuc of E dos ot dpd o th occuc of E d th occuc of E dos ot dpd o th occuc of E th E d E clld idpdt vts othwis th clld dpdt vts. A vt B c b plid b st of hustiv d mutull clusiv hpothsis A,A,.,A. Giv pioi pobbilitis PA,PA,,PA cospodig to totl bsc of kowldg gdig th occuc of B d coditiol pobbilitis PB/A,PB/A,,PB/A th postio pobbilit PA / B j of som vt Aj is giv b PA / B j PA.P B / A i j PA P B / A i j i th pobbilit PC/B of mtiliztio of oth vt C is giv b

66 66 PA / B j i PA.P B / A P C / A B i i i i PA P B / A i i A l vlud fuctio dfid o smpl-spc is clld domvibl. A dom vibl c ssum ol st of l vlus d th vlus which th vibl tks,dpds o th chc. Rdom vibl is lso clld stochstic vibl o simpl vit. Wh w dl with vits lik wights d tmptu th w kow tht ths vits c tk ifiit umb of vlus i giv itvl. Such tp of vits kow s cotiuous vits. Lt X b cotiuous dom vibl d lt th pobbilit of X fllig i th ifiitsiml itvl fd i.. P d X d f d, d, d b pssd b wh f is cotiuous fuctio of X d stisfis th followig two coditios: i f b if X b ii f d, f d, if X th th fuctio f is clld th pobbilit dsit fuctio Th pobbilit distibutio of cotiuous dom vit is clld th cotiuous pobbilit distibutio d it is pssd i tms of pobbilit dsit fuctio. Th pobbilit tht th vlu of dom vit X is o lss th is clld th cumultiv distibutio fuctios of X

67 67 Lt b th disct dom vibl d lt its fquc distibutio b s follows : Vit :.. Pobbilit : p p p.. p wh p p p... p p, th th mthmticl pcttio of o simpl pcttio of is dotd b E d is dfid b E p p p... p p p i i i Th pcttio of th sum of two vits is qul to th sum of thi pcttios, i.., if d two vits th E E E Th pcttio of th poduct of two idpdt vits is qul to th poduct of thi pcttios E E. E Lt d b two dom vibls d d b thi pctd vlus o ms spctivl. Th co-vic btw d, dotd b cov,, is dfid s cov, E[ ].

68 68.8 CHECK YOUR PROGRESS : THE KEY Ecis Q. A coi is tossd twic. Wht is th pobbilit tht t lst o hd occus. Q. A cd is dw fom odi dck. Fid th pobbilit tht it is ht. Q. Wht is th pobbilit of gttig totl of 7 o wh pi of dic is tossd? Q.4 A coi is tossd 6 tim si succssio. Wh is th pobbilit tht t lst o hd occus? Q.5 A cd is dw fom odi dck d w told tht it is d. Wht is th pobbilit tht th cd is gt th but lss th 9. Q.6 Fid fomul fo th pobbilit distibutio of th dom vibl X pstig th outcom wh sigl di is olld oc.

69 69 UNIT. THEORETICAL DISTRIBUTIONS Stuctu.. Itoductio.. Objctivs.. Thoticl Distibutios.4. Biomil Distibutio.4.. Costts of th Biomil Distibutio.4.. Rovsk Fomul: Rcuc ltio fo th momt of Biomil distibutio..4.. Momt Gtig Fuctio of Biomil Distibutio.4.4. Mod of Biomil Distibutio.4.5. Illusttiv Empls.5. Poisso s Distibutio.5.. Costts of th Poisso s Distibutio.5.. Rcuc ltio fo th momt of Poisso s distibutio..5.. Momt Gtig Fuctio of Poisso s Distibutio.5.4. Mod of Poisso s Distibutio.5.5. Illusttiv Empls.6. Noml Distibutio

70 7.6.. Costts d cuc ltio of th Noml Distibutio..6.. Momt Gtig Fuctio of Noml Distibutio.6.. Illusttiv Empls.7. Rctgul Distibutio.7.. Costts of th Rctgul Distibutio.7.. Momt Gtig Fuctio of Rctgul Distibutio.7.. Illusttiv Empls.8. Epotil Distibutio.8.. Costts of th Epotil Distibutio.8.. Momt Gtig Fuctio of Epotil Distibutio.8.. Illusttiv Empls.9. Summ of Uit.. Assigmt.. Chck ou pogss.. Poit of Discussio.. Suggstd Stud Mtil.. Itoductio: W ld fmili to th cocpts of fquc distibutio, msu of ctl tdc msu of dispsio m dvitio, stdd dvitio, momt, skwss, kutosis, tho of pobbilit, mthmticl pcttio d momt gtig fuctios. I this uit w shll cofi ov slvs to th stud of thoticl distibutio Biomil, Poisso, Noml, Rctgul d Epotil distibutios. Futh w dl with th poptis d pplictios of ths distibutios. Ths

71 7 distibutios dividd ito two pts Disct Thoticl Distibutios d Cotiuous Thoticl Distibutios. Biomil d Poisso distibutios Disct Thoticl Distibutios whil Noml, Rctgul d Epotil Cotiuous Thoticl Distibutios. If cti hpothsis is ssumd, it is somtims possibl to div mthmticll wht th fquc distibutios of cti uivss should b. Such distibutios clld Thoticl Distibutios. Th Biomil Distibutio ws discovd b Jms Boulli i 7 d thfo it is lso clld th Boulli Distibutio. Th Poisso distibutio is pticul limitig fom of th Biomil distibutio; it ws fist discovd b Fch Mthmtici S.D. Poisso i 87. I 7 Dmoiv md th discov of Noml o Gussi distibutio s limitig fom of Biomil distibutio, it is dfid b th pobbilit dsit fuctio. Rctgul o Uifom d Epotils distibutios lso dfid b difft pobbilit dsit fuctios. Ths distibutios sv s th guidig istumt i schs i th phsicl, socil scics d i mdici, gicultu d giig. Ths idispsbl tool fo th lsis d th itpttio of th bsic dt obtid b obsvtio Epimt..: Objctivs: Aft th d of th uit th studt will b bl to udstd/kow th. Cocpts of Thoticl Distibutios. Diffc btw th Disct d Cotiuous Thoticl distibutios. Biomil d Poisso Distibutios Poptis, costts d pplictios 4. Noml, Rctgul d Epotil Distibutios Poptis, costts d pplictios 5. Abl to solv th poblms bsd o bov distibutios 6. Abl to us of ths distibutios i v d lif clcultios 7. Mits d D-mits of ths distibutios.. Thoticl Distibutios: Wh th fquc distibutios md b collctig th dt i dict fom. Th such distibutios clld obsvd fquc distibutios. Wh th fquc distibutios md b obtiig pobbl o pctd fquc b usig mthmticl mthods o th bsis of dfiit hpothsis o ssumptios th such distibutios sid to b thoticl fquc distibutios. Thoticl distibutios obtid b pobbilit distibutios. If th pobbilitis

72 7 ssumd to th ltiv fqucis th th pobbilit distibutios sid to b thoticl fquc distibutios. Thoticl distibutios dividd ito two followig pts:. Disct thoticl fquc distibutios Biomil distibutio b Poisso distibutio. Cotiuous thoticl fquc distibutios Noml distibutio b Rctgul distibutio c Epotil distibutio Oth soms of thoticl fquc distibutios mthmticl fquc distibutios, idl distibutios, d pctd fquc distibutios. W c udstd th thoticl distibutio b followig mpl: Empl: If w thow fou cois 8 tims th b th tho of pobbilit th thoticl distibutio is giv s : No of tils Pobbilit Epctd fqucis = = = = 6 8 = 5 6 Totl 8

73 7.4. Biomil Distibutio: Lt th b vt, th pobbilit of its big succss is p d th pobbilit of its filu is q i o til, so tht p + q =. Lt th vt b tid tims d suppos tht th tils i Idpdt ii Numb of tils fiit d iii Th pobbilit p of succss is th sm i v til. Th umb of succsss i tils m b,,,...,,,. Th pobbilit tht th fist tils succsss d th miig filus is p q -. But w to cosid ll th css wh tils succsss, sic out of, c b chos C ws, th th pobbilit p [o b, p, of succsss out of idpdt tils is giv b p = C p q -. Thus th umb succss, c tk th vlus,,,,,, with cospodig pobbilitis q, C p q -, C p q -,, C p q -,, p. Th pobbilit of th umb of succsss so obtid is clld th biomil pobbilit distibutio fo th obvious so tht th pobbilitis th vis tms i th biomil psio of q + p. Th sum of th pobbilitis = b, p, C p q q p. Dfiitio: Th pobbilit distibutio of dom vibl is clld Biomil distibutio if ol tk o-gtiv vlus d its distibutio is giv b, P = px = = C p q ;,,,,..., ;,,,,..., Th biomil distibutio cotis two idpdt costts viz, d p o q, ths clld th pmt of th biomil distibutio. If p = q = ½, th biomil distibutio is clld smmtic d wh p q it is clld skw smmtic distibutio. Lt th idpdt tils costitut o pimt d lt this pimt ptd N tims wh N is v lg. I ths N sts th will b fw sts i which th is o succss, fw sts of o succss, fw sts of two succsss, d so o. Hc i ll th N sts, th umb of sts with succsss is N C p q -. Thfo th umb of sts cospodig to th umb of succsss,,,,,,, spctivl N q, N C p q -, N C p q -,, N C p q -,, N p.hc fo N sts of tils th Thoticl fquc distibutio o Epctd fqucis distibutio of,,,,,, succsss giv b th succssiv tms i th pssio