MA6451-PROBABILITY AND RANDOM PROCESSES

Transcription

1 MA645-PROBABILITY AND RANDOM PROCESSES UNIT I RANDOM VARIABLES Dr. V. Valliammal Darm of Alid Mahmaics Sri Vkaswara Collg of Egirig

2 Radom variabl Radom Variabls A ral variabl whos valu is drmid by h oucom of a radom rim is calld a radom variabl..g A radom rim cosiss of wo osss of a coi. Cosidr h radom rim which is h umbr of hads, or Oucom: HH HT TH TT Valu of :

3 O Dimsioal Radom Variabl ad Two dimsioal Radom Variabl A ral valud fucio dfid o S ad alkig valus i R, is calld a o-dimsioal radom variabl. If h valus ar ordrd airs of ral umbrs, h fucio is said o b wo-dimsioal radom variabl. No: I ui-i radom variabls ad hir robabiliy disribuios w rsricd ourslvs o o dimsioal saml sacs.

4 Discr Radom Variabl A radom variabl which aks a couabl umbr of ral valus is calld a discr radom variabl..g. umbr of lho calls r ui im. marks obaid i a s. umbr of riig misaks i ach ag of a book

5 Probabiliy Mass Fucio If is a discr radom variabl akig amos a couably ifii umbr of valus,,.., w associa a umbr P i P i P i, calld h robabiliy mass fucio of. Th fucio P i saisfis h followig codiios: i P i i,,, ii P i i

6 Coiuous Radom Variabl A radom variabl is said o b coiuous if i ca ak all ossibl valus bw crai limis..g.. Th lgh of a im durig which a vacuum ub isalld is a coiuous radom variabl.. umbr of scrachs o a surfac, roorio of dfciv ars amog sd,. umbr of rasmid i rror.

7 Probabiliy Dsiy Fucio Cosidr a small irval, d of lgh d. Th fucio fd rrss h robabiliy ha falls i h irval, d i.., P d f d. Th robabiliy fucio of a coiuous radom variabl is calld as robabiliy dsiy fucio ad i saisfis h followig codiios. i f fd ii

8 Disribuio Fucio Th disribuio fucio of a radom variabl is dod as F ad is dfid as F P. Th fucio is also calld as h cumulaiv robabiliy fucio. F P P Fd wh is discr wh is coiuous

9 Proris o Cumulaiv Disribuio. If a b, Fa Fb, whr a ad b ar ral quaiis.. If F is h disribuio fucio of a odimsioal radom variabl, h F.. If F is h disribuio fucio of a o dimsioal radom variabl, h F ad F.

10 Problms. If a radom variabl aks h valus,,, 4 such ha PPP5P4. Fid h robabiliy disribuio of. Soluio: Assum P α By h giv quaio α α P P P 4 For a robabiliy disribuio ad mass fucio Σ P PPPP4 α 5

11 α α α 6 α α α 5 5 P ; P ; P ; P Th robabiliy disribuio is giv by

12 . L b a coiuous radom variabl havig h robabiliy dsiy fucio Fid h disribuio fucio of. Soluio: f,, ohrwis F f d d

13 . A radom variabl has h robabiliy dsiy fucio f giv by Fid h valu of c ad CDF of. Soluio: ohrwis c f,, c c c d c d f d d c d f F

14 4. A coiuous radom variabl has h robabiliy dsiy fucio f giv by Fid h valu of c ad CDF of. Soluio: f f d c d c d c d c c c c, < <

15 c d c d c d f F i Cas < > c c c c c c d c d c d c d f F ii Cas < >,, F

16 5. A radom variabl has h followig robabiliy disribuio. : f: k k k k k k 7k k Fid i h valu of k ii.5 < < 4.5 > ad iii h smalls valu of such ha > /. Soluio i k k k P k 9k. k k k k k, 7k k

17 ii A.5 < B A B.5 < < 4.5 {,,4} {,4,5,6,7} {,4} > < 4.5 > A B A B,4 B,4,5,6,7 5 k k 5k 7 k k k k 7k k k 6k iii F k.. k..5 4 k..8 5 k..8 6 k k k.7. From h abl for 4,5,6,7 > ad h smalls valu is 4 Thrfor

18 Ecaio of a Radom Variabl Th caio of a radom variabl is dod as E. I rurs a rrsaiv valu for a robabiliy disribuio. For a discr robabiliy disribuio E. For a coiuous radom variabl which assums valus i a, b E b a fd

19 Proris o Ecaio. Ecaio of a cosa is a cosa.. E[a] ae, whr a is a cosa.. Ea b ae b, whr a ad b ar cosas. 4. E E, for ay radom variabl. 5. If Y, E EY.

20 Variac of a Radom Variabl Th variac of a Radom variabl, which is rrsd as V is dfid as h caio of squars of h drivaios from h cd valu. V E E Proris O Variac. Variac of a cosa is. Vara b a Var, whr a is a cosa.

21 Moms ad Ohr Saisical Cosas Raw Moms Raw moms abou origi Raw moms abou ay arbirary valu A Cral moms µ r µ r b b a a r fd A r fd µ r E[ E] r b a E r fd

22 Rlaioshi bw Raw Moms ad Cral Moms µ always µ µ µ µ µ µ µ µ µ 4 4 µ 4 4µ µ 6µ µ µ

23 Mom Graig Fucio M.G.F I is a fucio which auomaically gras h raw moms. For a radom variabl, h mom graig fucio is dod as M ad is drivd as M E. Raso for h am M.G.F M E E!! E E E!

24 E E! µ µ! Hr µ coffici of i M µ coffici of i M! I gral µ r coffici of i M.!

25 Problms. Th.m.f of a RV, is giv by Fid MGF, ma ad variac. Soluio... 4 E M.. 4

26 Diffriaig wic wih rsc o u abov M 4 4 M M E E E Variac M E

27 . Fid MGF of h RV, whos df is giv by ad hc fid h firs four cral moms. Soluio M E f d d d

28 Eadig i owrs of M Takig h coffici w g h raw moms abou origi E coffici of! E coffici of! E E coffici of! 4 4 coffici of 4!

29 ad h cral moms ar µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ C C C C C C

30 . If h MGF of a discr RV is fid h disribuio of ad 5 or 6. Soluio By dfiiio M... E M

31 Sadard Disribuios Biomial Disribuio Assumios.Th radom rim corrsods o oly wo ossibly oucoms..th umbr of rials is fii..th rials ar idd. 4.Th robabiliy of succss is a cosa from rial o rial.

32 Noaios umbr of rials robabiliy of succss q robabiliy of failur A radom variabl which rrss h umbr of succsss Biomial Disribuio A discr radom variabl is said o follow Biomial disribuio if i s robabiliy mass fucio is P C q,,,,...,

33 Mom Graig Fucio Th M.G.F of a Biomial varia is P ] E[ M q q C q C M

34 Raw Moms µ d d M d d q q q [ q ] Hc ma of Biomial disribuio

35 Variac of Biomial Disribuio q µ µ µ ] q [ ] [ ] q q [ ] q q [ ] q [ d d M d d µ

36 Addiiv Prory of Biomial Disribuio L follow biomial disribuio wih aramrs ad. L follow Biomial disribuio wih aramrs ad. Furhr l ad b idd. Cosidr M M M q q M M [ q Which is o of h form q. Hc is o a biomial varia. q ad ar idd]

37 Bu if, h q q q. Equaio bcoms M q q q Which is of h form q. Hc follows Biomial disribuio wh. i.., Biomial disribuio has addiiv rory wh.

38 Problms Basd O Biomial Disribuio. I has b claimd ha i 6 % of all solar ha isallaio h uiliy bill is rducd by alas ohird. Accordigly wha ar h robabiliis ha h uiliy bill will b rducd by alas o-hird i alas four of fiv isallaio. Soluio Giv 5, 6%.6 ad q c [ 4.7 4].6 4 [ ] 5c

39 . Th ma ad variac of a biomial varia ar 6 ad rscivly. Fid P. Soluio Giv ha E 6 i.., 6 ad V q Dividig Equaio by Equaio givs, q 6 q q

40 Us i Equaio C q C P C P P P 9 9 <

41 . A ubiasd di is rolld ims. Gig a oucom grar ha 4 i a di is rmd as a succss. Wha is h chac of gig a las 8 succsss? Soluio : umbr of succsss : P umbr of succss P oucom is grar C ha 4 i a di 6 C q umbr of rials

42 P alas 8 succsss P 8,,,..., C q C P C C C C P

43 Poisso Disribuio Th alicaio of Biomial disribuio will b ivalid wh ad. Hc i h rsc of h abov wo codiios w d a horical disribuio which ovrcoms h drawback of Biomial disribuio. Th Biomial disribuio ds o oisso disribuio wh i.th umbr of rials is idfiily larg i.., ii.th robabiliy of succss is vry small i.., iii. is a cosa i..,.

44 Poisso Disribuio mf A discr radom variabl is said o follow Poisso disribuio wih aramr if is robabiliy mass fucio is P Mom Graig Fucio!,,,,..., M E[ ] P!! a! a

45 Raw Moms of Poisso Disribuio i.., ma of Poisso disribuio. d d M d d µ

46 Scod Ordr Raw Mom ] [ ] [ ] [ d d d dm. d d d M d µ

47 Variac of Poisso disribuio Variac Scod ordr cral mom µ µ µ Thus i Poisso disribuio ma Variac.

48 Addiiv Prory of Poisso Disribuio L ad b wo idd Poisso varias wih aramr ad rscivly. Th M ad M Cosidr h varia Now M M...M [ Which is of h form. Hc f ollows Poisso disribuio wih aramr. i.., h Poisso disribuio has addiiv rory. ad ar idd]

49 Prov ha oisso disribuio is h limiig cas of Biomial disribuio. or Poisso disribuio is a limiig cas of Biomial disribuio udr h followig codiios i, h o. of rials is idfiily larg, i., ii, h cosa robabiliy of succss i ach rial is vry small, i. iii is if ii or ad q, is osiiv ral Soluio If is biomial r.v wih aramr &, h

50 q c...!....!!...!!!...!!!,,,...,

51 Takig limi as o boh sids ad i is oisso disb. Hc h roof.,...,,,!....!...!...! lim lim lim lim lim,,,...,!

52 Prov ha h sum of wo idd oisso varias is a oisso varia, whil h diffrc is o a oisso varia. Soluio L ad b idd r.v.s ha follow oisso disb. wih Paramrs ad rscivly. L

53 This is oisso wih aramr [ ] [ ] r r r r r r r r r r r r r r c r r r r r r aridd c r r!.!!!!..!!!.!.!.!..!. & si.

54 ii Diffrc is o oisso L I is o oisso. [ ] E E E E [ ] [ ] [ ] [ ] [ ] [ ] E E E E E E E

55 Prov ha oisso disribuio is h limiig cas of Biomial disribuio. or Poisso disribuio is a limiig cas of Biomial disribuio udr h followig codiios i, h o. of rials is idfiily larg, i., ii, h cosa robabiliy of succss i ach rial is vry small, i. iii is if ii or ad q, is osiiv Soluio If is biomial r.v wih aramr &, h ral

56 q c...!....!!...!!!...!!!,,,...,

57 Takig limi as o boh sids ad i is oisso disb. Hc h roof.,...,,,!....!...!...! lim lim lim lim lim,,,...,!

58 Problms Basd o Poisso Disribuio. Th o. of mohly brakdows of a comur is a r.v. havig oisso disb wih ma.8. Fid h robabiliy ha his comur will fucio for a moh wih oly o brakdow. Soluio, giv.8!.8.8!.975

59 . I is kow ha h robabiliy of a im roducd by a crai machi will b dfciv is.5. If h roducd ims ar s o h mark i acks of, fid h o. of acks coaiig alas, acly, amos dfcivs i a cosigm of acks usig oisso. Soluio Giv,.5, N Ma L do h o. of dfcivs.

60 [ ].! [ ] [ < ] [ ]!!.64 Thrfor, ou of acks, h o. of acks coaiig alas dfcivs ii [ ]. 895! Ou of acks,n*[]84 acks iii [ ] [ ] [ ] [ ] For acks * acks aroimaly..!.!,,,... [ ] *.64 acks N. 64!!!.9975

61 . Th aoms of radio aciv lm ar radomly disigraig. If vry gram of his lm, o avrag, mis.9 alha aricls r scod, wha is h robabiliy durig h scod h o. of alha aricls mid from gram is i amos 6 ii alas iii alas ad amos 6? Soluio Giv.9 L do h o. of alha aricls mid i !.9.9!.9.9! ! 6.898

62 [ ].9!.9! < ii.645 6!.9 5!.9 4!.9! iii

63 Gomric Disribuio A discr radom variabl which rrss h umbr of failurs rcdig h firs, succss is said o follow Gomric disribuio if is robabiliy mass fucio is P q,,,, Mom Graig Fucio M E[ ] P q q [ q [ q ] q ]

64 Raw Moms of Gomric Disribuio q q q q q q q q q d d M d d µ

65 q q q q ] q q q[q ] q q q q[ µ q q d d d M d µ

66 Variac of Gomric Disribuio ] q [ q q q q q q q q q q µ µ µ

67 Ecaio of Gomric Disribuio wihou usig MGF q q q] q[...] q q q[...] q q [ q P E

68 Variac of Gomric Disribuio [E] E V q q q q q] [ q...] 4q q [ q...] 8q 6q q [ q P P ]P [ P ] E[ q q q 4 q

69 ] q [ q q q q q q q q q q [E] E V

70 Esablish h mmorylss rory of gomric disb. Soluio If is a discr r.v. followig a gomric disb. Now [ ] [ ] k k k k k k k k q q q q q q q q q q q k q >......,,..., [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] m m q q q m m m m ad m m m m m > > > > > > > > > > >

71 Problm Basd o Gomric Disribuio. Suos ha a rai soldir shoos a arg i a idd fashio. If h robabiliy ha h arg is sho o ay o sho is.7. i Wha is h robabiliy ha h arg would b hi i h am? ii Wha is h robabiliy ha i aks him lss ha 4 shos? iii Wha is h robabiliy ha i aks him a v o. of shos? iv Wha is h avrag o. of shos dd o hi h arg? Soluio L do h o. of shos dd o hi h arg ad follows gomric disribuio wih mf [ ] q,,,...

72 i ii iii iv [ ] [ < 4] [ is avumbr] [.....].. [. ] [ ] Avrag o. of shos E.486.7

73 Uiform Disribuio Coiuous Uiform Disribuio or Rcagular Disribuio A coiuous radom variabl dfid i h irval a,b is said o follow uiform disribuio if is robabiliy dsiy fucio is giv by f b, a, a < < b ohrwis No: ad b ar h aramrs of h disribuio.

74 Mom Graig Fucio of a Uiform Disribuio MGF a b a b d a b fd E M a b b a b a b a

75 Raw Moms of Rcagular Disribuio Wh r, Equaio bcoms Ma r a b a b µ r a b d a b fd µ r r r b a r b a r b a r r a b a b a ab b a b a b µ

76 Wh r, Equaio bcoms Variac of Uiform Disribuio a ab b a ab ab b a b a b a b µ b a ab b a a 6ab b 4a 4ab 4b a b a ab b µ µ [E] E V

77 Problms Basd o Uiform Disribuio.Show ha for h uiform disribuio f, a < < a Soluio: Giv a sih a, h mgf abou origi is. f, a < < a a a MGF M E [ ] M a a f d a a sih a a d a a [ ] a a d a a a a sih a a sih a a

78 . Th umbr of rsol comur c sold daily a a comur world is uiformly disribud wih a miimum of c ad a maimum of 5 c. Fid Th robabiliy ha daily sals will fall bw 5 ad c Wha is h robabiliy ha h comur world will sll alas 4 c s? Wha is h robabiliy ha h comur world will sll acly 5 c s? Soluio L ~Ua, b, h h df is giv by

79 f, a < < b b a, < < 5 5, < < 5 [ 5 < < ] [ 4] f d [ ] [ 5]. 66 [ 5 4]. i. i is aricular oi, h valu is zro. [ 5 ] f d 5 4 d d [ ] 5 5 4

80 . Sarig a 5. am vry half a hour hr is a fligh from Sa Frasisco airor o Losagls. Suos ha o of hr las is comlly sold ou ad ha hy always hav room for assgrs. A rso who was o fly o Losagls arriv a a radom im bw 8.45 am ad 9.45 am. Fid h robabiliy ha sh wais a Amos mi b alas 5 mi Soluio L b h uiform r.v. ovr h irval, 6. Th h df is giv by f b a, 6, a < < < b < 6

81 a Th assgrs will hav o wai lss ha mi. if sh arrivs a h airor 5 < < 5 5 < [ ] 5 [ ] b Th robabiliy ha sh has o wai alas 5 mi. 5 < < d 5 6 < < d d [ ] [ ] 5 < 6 45 d 6 45

82 Eoial Disribuio A coiuous radom variabl is said o follow oial disribuio if is df is giv by Ma ad variac of a Eoial disribuio:, f [ ] d d d f E Ma

83 Ma [ ] d d d f E [ ] E E Variac

84 Mom Graig Fucio of a Eoial Disribuio ] [ d d fd ] E[ M

85 Esablish h mmory lss rory of a oial disribuio. Soluio If is oially disribud, h Th df of oial dis is giv by [ ] [ ], > > > > for ay s s s ohrwis f,, [ ] > k k k k d d f k [ ] [ ] [ ] [ ] [ ] [ ] s s s s ad s s s s s > > > > > > > > [ ] [ ], > > > > for ay s s s

86 Problms basd o Eoial Disribuio Th im i hours rquird o rair a machi is oially disribud wih aramr /. awha is h robabiliy ha h rair im cds hrs? bwha is h codiioal robabiliy ha a rair aks alas hrs giv ha Is dircio cds 8 hrs? Soluio If rrss h im o rair h machi, h dsiy fucio Of is giv by f,

87 [ ] 679. > d d d f [ ] [ ]. 8 > > d d d f

88 Gamma Disribuio A coiuous radom variabl is said o follow Gamma disribuio wih aramr if is robabiliy dsiy fucio is f Γ ohrwis Mom graig fucio of gamma disribuio M E[ ] fd d d

89 Raw moms of gamma disribuio Variac of gamma disribuio Hc i gamma disribuio ma variac d d M d d µ d d d M d µ µ µ µ

90 Variac of gamma disribuio V E [E] Addiiv rory of Gamma Disribuio Cosidr wo idd gamma varias ad wih aramrs ad rscivly. M M M M M Which is of h form. Hc is also a Gamma varia wih aramr. Hc Gamma disribuio has addiiv rory. [ ad ar idd]

91 Problms Basd o Gamma Disribuio. I a crai ciy h daily cosumio of lcric owr i millios of kilowa hrs ca b rad as cral gamma dis wih, k. If h owr la has a daily caaciy of millio kilowa hours. Wha is h robabiliy ha h owr suly will b iadqua o ay giv day. Soluio L b h daily cosumio of lcric owr Th h dsiy fucio of is giv by

92 [ h owr suly is iadqua][>] 6! 8 k k k f Γ Γ d d d f

93 . Th daily cosumio of milk i a ciy i css of, lirs is aroimaly disribud as a Gamma dis wih aramr, k. Th ciy has a daily sock of, lirs. Wha is h robabiliy ha h sock is isuffici o a aricular day. Soluio L b h daily cosumio, so, h r.v. Y-. Th k k y f Y y y Γ k y y y Γ y y y!

94 [ isuffici sock][>] [Y>].757,, ] [ > y u By subsiuio mhod dy y dy y dy y f Y y y

95 Normal Disribuio Th Eglish mahmaicia D-Moivr, obaid a coiuous disribuio as a limiig cas of biomial disribuio i h yar 7. This disribuio was amd ormal disribuio. Th firs rso who mad rfrc o his disribuio was Gauss who usd i o sudy h disribuio of rrors i Asroomy. Probabiliy Dsiy Fucio A coiuous radom variabl is said o follow ormal disribuio wih aramrs µ ma ad σ variac, i is dsiy fucio is giv by h robabiliy law: f σ µ σ, < <, < µ < adσ >

96 Rmarks A radom variabl which follows Normal disribuio wih ma µ ad variac σ is rrsd as ~ Nµ, σ. µ If is a ormal varia, z is calld as a sadard σ ormal varia. If ~ Nµ, σ, h z ~ N,. Th df of a sadard ormal varia Z is giv by, φz π Z /, < Z <

97 Mom Graig Fucio of Normal Disribuio If follows ormal disribuio wih ma µ ad variac σ, is mom graig fucio is drivd as follows: M E[ ] fd σ π µ σ d L µ z d σdz σ z z

98 dz π dz π dz π σdz π σ M σz z µ z σz µ z σz µ z σz µ dz π dz π dz π M σσ z σ µ σ σ σz z µ σ σ σz z µ

99 z σ A dz da z A z A M µ π σ A da A π da bcaus A π is h df of a sadard ormal varia. M µ σ

100 Graig h raw moms usig MGF µ µ σ d dm µ M µ σ σ µ σ µ σ µ σ µ µ σ µ σ σ µ σ µ µ σ σ µ µ µ σ σ d M d µ

101 Problms basd o Normal Disribuio. Fid P z > -.9 Soluio >. 5 < Z.9 P z > -.9 P-.9 < z <.5 P < z <

102 . Fid P - z - Soluio Z- Z- P - z - P z - P z

103 . % of h obsrvaios i h ormal disribuio ar blow 6. 8% of h obsrvaios ar blow. Comu h ma ad sadard dviaio of h disribuio. Soluio L b h ormal varia Giv ha P 6. 6 µ P z σ Also giv ha P.8 µ P z σ..8 Rrsig Equaios ad i h diagram

104 . 6 µ σ From h diagram... µ σ µ P < z <. σ µ.85 [From h abls] σ µ.85σ

105 Agai from h diagram, Addig Equaios ad 4 6 µ P < z <. σ µ 6 P < z <. [By symmry] σ µ 6.85 [From abls] σ µ 6.85σ 6.7σ σ Puig Hc σ 5.9 i µ µ 9 ad µ 9 ad σ σ

106 Summarizig h Ui Radom Variabls:. Discr R.V. Coious R.V Sadard Disribuio Udr Discr -. Biomial Dis. Poisso Dis. Gomric Dis Udr Coiuous-. Uiform Dis. Eoial Dis. Gamma Dis 4. Normal Dis

107 FORMULAE Sl. No. Discr radom variabl Coiuous radom variabl. i i fd. F P[ ] F P[ ] fd. Ma E[] i i i Ma E[] fd E[ ] i Var E i i [E] E[ ] fd Var E [E] Mom E[ M.G.F. M E[ r ] i r ] i i r r Mom E[ ] fd M.G.F M E[ ] fd

108 4 Ea b ae b 5 Vara b a Var 6 Vara ± by a Var b VarY 7 Sadard Dviaio 8 f F 9 > a a A B A/B, B B Var If A ad B ar idd, h A B A.B.

109 .P.D.F, M.G.F, Ma ad Variac of all h disribuios: Sl. No. Disribuio P.D.F. P M.G.F Ma Variac. Biomial C q q q. Poisso!

110 Sl. No. Disribuio P.D.F. P M.G.F Ma Variac. Gomric q or q q q 4 uiform 5 Eoial, f b a, a < < b ohrwis f, >, >, ohrwis b a b a a b 6 Gamma f Γ f, < <, > b a. Thak you