Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 2 Solutions

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 2 Solutions

1/15

@0/2

B0/2

N0/=

]8/=

N8/=

]0/=

@N 0/>

@] 8/>

BN 8/>

B] 0/>

]nj preb`bflfty tn`t yeu iujss herrjhtly fs

T SHW 7 T S@]W + T SBNW 7 8/> + 8/> 7 8/=. (

Trebljg 2.0.: Xelutfek

]nj TS |NW fs tnj preb`bflfty tn`t ` pjrsek wne n`s NFQ tjsts kji`tfvj atnj mfsj`sj. ]nfs fs rjajrrjm te `s ` a`lsj-kji`tfvj rjsult. ]nj h`sj wnj` pjrsek wne mejs ket n`vj NFQ but tjsts pesftfvj aer tnj mfsj`sj, fs h`llj` a`lsj-pesftfvj rjsult `km n`s preb`bflfty TS+|NhW. Xfkhj tnj tjst fs herrj66% ea tnj tfgj,

T S|NW 7 T S+|NhW 7 4.40. (

Kew tnj preb`bflfty tn`t ` pjrsek wne n`s tjstjm pesftfvj aer NFQ `htu`ln`s tnj mfsj`sj fs

T SN|+W 7T S+, NWTS+W

7 T S+, NWT S+, NW + T S+, NhW

. (

Yj h`k usj B`yjs aergul` te jv`lu`tj tnjsj oefkt preb`bflftfjs.

T SN|+W 7 T S+|NW T SNW

T S+|NW T SNW + T S+|NhW T SNhW

7 (4.66)(4.4442)

(4.66)(4.4442) + (4.40)(4.666>)7 4.406=. (

]nus, jvjk tneuin tnj tjst fs herrjht 66% ea tnj tfgj, tnj preb`bflfty tn` r`kmeg pjrsek wne tjsts pesftfvj `htu`lly n`s NFQ fs ljss tn`k 4.42. ]rj`sek tnfs preb`bflfty fs se lew fs tn`t tnj ` prferf preb`bflfty tn`t ` pjrsen`s NFQ fs vjry sg`ll.

89


2/15

Trebljg 2.0.04 Xelutfek

(`) Yj wfsn te ckew wn`t tnj preb`bflfty tn`t wj ffkm ke ieem pnetemfemfkkp`frs ea mfemjs. ]jstfki j`hn p`fr ea mfemjs fs `k fkmjpjkmjkt trfsuhn tn`t wftn preb`bflfty p, betn mfemjs ea ` p`fr `rj b`m. Are

Trebljg 2.0.1, wj h`k j`sfly h`lhul`tj p.

p7 T Sbetn mfemjs `rj mjajhtfvjW 7 T SM0M2W 7 1/2:. (

]nj preb`bflfty ea_k, tnj preb`bflfty ea zjre `hhjpt`blj mfemjs eut k p`frs ea mfemjs fs pk bjh`usj ek j`hn tjst ea ` p`fr ea mfemjs, begust bj mjajhtfvj.

T S_kW 7

kf70

p7 pk 7 1

2:k

(

(b) @ketnjr w`y te pnr`sj tnfs qujstfek fs te `sc new g`ky p`frs gust wtjst uktfl TS_kW 4.40. Xfkhj TS_kW 7 (1/2:)

k, wj rjqufrj1

2:

k4.40 k

lk 4.40

lk 1/2:7 8.28. (

Xfkhj kgust bj `k fktjijr, k7 = p`frs gust bj tjstjm.


]nj st`rtfki pefkt fs te mr`w ` trjj ea tnj jxpjrfgjkt. Yj mjffkj tnj jvjkY tn`t tnj pl`kt fs w`tjrjm, L tn`t tnj pl`kt lfvjs, `km M tn`t tnj pl`mfjs. ]nj trjj mf`ir`g fs

Y4.9

Yh4.8

L4.>

M4.2

L4.0 M4.6

YL 4.:1

YM 4.0=

YhL 4.48

YhM 4.29

=8


3/15

Ft aellews tn`t

(`) TSLW 7 TSY LW + TSYhLW 7 4.:1 + 4.48 7 4.:6.

(b)

T SYh|MW 7T SYh

MWT SMW 7 4.294.0= + 4.29729=0 . (

(h) TSM|YhW 7 4.6.

Fk fkaerg`l hekvjrs`tfek, ft h`k bj hekausfki te mfstfkiufsn bjtwjjk TSM|Y`km TSYh|MW3 newjvjr, tnjy `rj sfgplj ekhj yeu mr`w tnj trjj.


]nj jxpjrfgjkt jkms `s seek `s ` ffsn fs hùint. ]nj trjj rjsjgbljs

H0p

Hh00p

H2p

Hh20p

H8p

Hh80p

...

Areg tnj trjj, TSH0W 7 p `km TSH2W 7 (0 p)p. Afk`lly, ` ffsn fs hùint etnj ktn h`st fa ke ffsn wjrj hùint ek tnj prjvfeus k 0 h`sts. ]nus,

T SHkW 7 (0 p)k0p. (

Trebljg 2.2.0 Xelutfek]jhnkfh`lly, ` iugb`ll g`hnfkj n`s ` ffkftj kugbjr ea iugb`lls, but tprebljg mjshrfptfek gemjls tnj mr`wfki ea iugb`lls `s s`gplfki areg tng`hnfkj wftneut rjpl`hjgjkt. ]nfs fs ` rj`sek`blj gemjl wnjk tnj g`hnfkn`s ` vjry l`rij iugb`ll h`p`hfty `km wj n`vj ke ckewljmij bjaerjn`km

==


4/15

new g`ky iugb`lls ea j`hn heler `rj fk tnj g`hnfkj. Zkmjr tnfs gemjl, tnrjqujstjm preb`bflfty fs ifvjk by tnj gultfkegf`l preb`bflfty

T SR2V2I2B2W 7 >!

2!2!2!2!

0

=

20

=

20

=

20

=

2

7

>!

=04 4.48>:. (


Fk tnfs gemjl ea ` st`rburst p`hc`ij, tnj pfjhjs fk ` p`hc`ij `rj helljhtjm bs`gplfki wftneut rjpl`hjgjkt areg ` if`kt helljhtfek ea st`rburst pfjhjs.

(`) J`hn pfjhj fs bjrry er hnjrry wftn preb`bflftyp 7 0/2. ]nj preb

bflfty ea ekly bjrry er hnjrry pfjhjs fs p02 7 0/=461.

(b) J`hn pfjhj fs ket hnjrry wftn preb`bflfty 8/=. ]nj preb`bflfty `ll pfjhjs `rj ket pfkc fs (8/=)02 7 4.4809.

(h) Aerf7 0, 2, . . . , 1, ljtHf mjketj tnj jvjkt tn`t `ll 02 pfjhjs `rj fl`verXfkhj j`hn pfjhj fs fl`ver f wftn preb`bflfty 0/=, TSHfW 7 (0/=)

02. XfkHf `km Ho `rj gutu`lly jxhlusfvj,

TSA0W 7 TSH0 H2 H=W 7=f70

TSHfW 7 = TSH0W 7 (0/=)00.


(`) Ljt Bf, Lf, Ef `km Hf mjketj tnj jvjkts tn`t tnj ftn pfjhj fs BjrrLjgek, Er`kij, `km Hnjrry rjspjhtfvjly. Ljt A0 mjketj tnj jvjkt tn`ll tnrjj pfjhjs mr`w `rj tnj s`gj fl`ver. ]nus,

A07{X0X2X8, L0L2L8, E0E2E8, H0H2H8} (

T SA0W 7 T SX0X2X8W + T SL0L2L8W + T SE0E2E8W + T SH0H2H8W (

=:


5/15

Trebljg 2.2.= Xelutfek

(`) Xfkhj tnjrj `rj ekly tnrjj pfjhjs ea j`hn fl`ver, wj h`kket mr`w aeupfjhjs ea `ll tnj s`gj fl`ver. Njkhj TSA0W 7 4.

(b) Ljt Mf mjketj tnj jvjkt tn`t tnj ftn pfjhj fs ` mfffljrjkt fl`ver are`ll tnj prfer pfjhjs. Ljt Xf mjketj tnj jvjkt tn`t pfjhj f fs tnj s`gfl`ver `s ` prjvfeus pfjhj. @ trjj aer tnfs jxpjrfgjkt fs rjl`tfvjly sfgpbjh`usj wj step tnj jxpjrfgjkt `s seek `s wj mr`w ` pfjhj tn`t fs ts`gj `s ` prjvfeus pfjhj. ]nj trjj fs

M0

0

X22/00

M2

6/00

X8=/04

M8

1/04

X=1/6

M=

8/6

Ketj tn`t?

TSM0W 7 0 bjh`usj tnj ffrst pfjhj fs mfffljrjkt sfkhj tnjrj n`vjkbjjk `ky prfer pfjhjs.

Aer tnj sjhekm pfjhj, tnjrj `rj 00 pfjhjs ljat `km 6 ea tnesj pfjh`rj mfffljrjkt areg tnj ffrst pfjhj mr`wk.

Ifvjk tnj ffrst twe pfjhjs `rj mfffljrjkt, tnjrj `rj 2 helers, j`hn wft8 pfjhjs (1 pfjhjs) eut ea 04 rjg`fkfki pfjhjs tn`t `rj ` mfffljrjfl`ver areg tnj ffrst twe pfjhjs. ]nus TSM8|M2M0W 7 1/04.

Afk`lly, ifvjk tnj ffrst tnrjj pfjhjs `rj mfffljrjkt fl`vers, tnjrj `8 pfjhjs rjg`fkfki tn`t `rj ` mfffljrjkt fl`ver areg tnj pfjhjs pr

vfeusly pfhcjm.

]nus TSM=|M8M2M0W 7 8/6. Ft aellews tn`t tnj tnrjj pfjhjs `rj mfffljrjwftn preb`bflfty

T SM0M2M8M=W 7 0

6

00

1

04

8

67

6

::. (

=9


6/15

@k `ltjrk`tj `ppre`hn te tnfs prebljg fs te `ssugj tn`t j`hn pfjhjmfstfkiufsn`blj, s`y by kugbjrfki tnj pfjhjs 0, 2, 8 fk j`hn fl`ver. `mmftfek, wj mjffkj tnj euthegj ea tnj jxpjrfgjkt te bj ` =-pjrgut`tfeea tnj 02 mfstfkiufsn`blj pfjhjs. Zkmjr tnfs gemjl, tnjrj `rj (02)=7

jqu`lly lfcjly euthegjs fk tnj s`gplj sp`hj. ]nj kugbjr ea euthegfk wnfhn `ll aeur pfjhjs `rj mfffljrjkt fsk=7 02 6 1 8 sfkhj tnjrj `rj

hnefhjs aer tnj ffrst pfjhj mr`wk, 6 hnefhjs aer tnj sjhekm pfjhj areg tntnrjj rjg`fkfki fl`vers, 1 hnefhjs aer tnj tnfrm pfjhj `km tnrjj hnefhaer tnj l`st pfjhj. Xfkhj `ll euthegjs `rj jqu`lly lfcjly,

T SA=W 7 k=(02)=

7 02 6 1 8

02 00 04 67

6

:: (

(h) ]nj sjhekm gemjl ea mfstfkiufsn`blj st`rburst pfjhjs g`cjs ft j`sfjr selvj tnfs l`st qujstfek. Fk tnfs h`sj, ljt tnj euthegj ea tnj jxpjrfgjbj tnj

02=

7 =6: hegbfk`tfeks er pfjhjs. Fk tnfs h`sj, wj `rj fikerfk

tnj ermjr fk wnfhn tnj pfjhjs wjrj sjljhtjm. Kew wj heukt tnj kugbea hegbfk`tfeks fk wnfhn wj n`vj twe pfjhjs ea j`hn ea twe fl`vers. Yh`k me tnfs wftn tnj aellewfki stjps?

0. Hneesj twe ea tnj aeur fl`vers.2. Hneesj 2 eut ea 8 pfjhjs ea ekj ea tnj twe hnesjk fl`vers.

8. Hneesj 2 eut ea 8 pfjhjs ea tnj etnjr ea tnj twe hnesjk fl`vers.

Ljt kf jqu`l tnj kugbjr ea w`ys te jxjhutj stjp f. Yj sjj tn`t

k07 =

27 1, k27 8

27 8, k87 8

27 8. (]nj kugbjr ea pessfblj w`ys te jxjhutj tnfs sjqujkhj ea stjps fs k0k2k8:= Xfkhj `ll hegbfk`tfeks `rj jqu`lly lfcjly,

T SA2W 7k0k2k8

02=

7 :==6:

7 1

::. (

=>


7/15


Xfkhj tnjrj `rj N 7:29

jqufpreb`blj sjvjk-h`rm n`kms, j`hn preb`bflfty

oust tnj kugbjr ea n`kms ea j`hn typj mfvfmjm byN.

(`) Xfkhj tnjrj `rj 21 rjm h`rms, tnjrj `rj

219

sjvjk-h`rm n`kms wftn `

rjm h`rms. ]nj preb`bflfty ea ` sjvjk-h`rm n`km ea `ll rjm h`rms fs

T SR9W 7

219

:29

721! =:!:2! 06!

7 4.44=6. (

(b) ]njrj `rj 02 a`hj h`rms fk ` :2 h`rm mjhc `km tnjrj `rj029

sjvjk h`

n`kms wftn `ll a`hj h`rms. ]nj preb`bflfty ea mr`wfki ekly a`hj h`rms

T SAW 7029 :29

7

02! =:!

:!:2! 7 :.62 04

1

. (

(h) ]njrj `rj 1 rjm a`hj h`rms (O,P,Cea mf`gekms `km nj`rts) fk ` :2 h`mjhc. ]nus ft fs fgpessfblj te ijt ` sjvjk-h`rm n`km ea rjm a`hj h`rmTSR9AW 7 4.


]njrj `rj N: 7:2:

jqu`lly lfcjly ffvj-h`rm n`kms. Mfvfmfki tnj kugbjr

n`kms ea ` p`rtfhul`r typj by Nwfll yfjlm tnj preb`bflfty ea ` n`km ea tntypj.

(`) ]njrj `rj21:

ffvj-h`rm n`kms ea `ll rjm h`rms. ]nus tnj preb`bflf

ijttfki ` ffvj-h`rm n`km ea `ll rjm h`rms fs

T SR:W 721:

:2:

7

21! =9!

20! :2! 7 4.42:8. (

Ketj tn`t tnfs h`k bj rjwrfttjk `s

TSR:W 721

:2

2:

:0

2=

:4

28

=6

22

=>,

wnfhn snews tnj suhhjssfvj preb`bflftfjs ea rjhjfvfki ` rjm h`rm.

=6


8/15

(b) ]nj aellewfki sjqujkhj ea subjxpjrfgjkts wfll ijkjr`tj `ll pessfblj auneusj

0. Hneesj ` cfkm aer tnrjj-ea-`-cfkm.

2. Hneesj ` cfkm aer twe-ea-`-cfkm.

8. Hneesj tnrjj ea tnj aeur h`rms ea tnj tnrjj-ea-`-cfkm cfkm.=. Hneesj twe ea tnj aeur h`rms ea tnj twe-ea-`-cfkm cfkm.

]nj kugbjr ea w`ys ea pjraergfki subjxpjrfgjkt f fs

k07

08

0

, k27

02

0

, k87

=

8

, k=7

=

2

. (

Ketj tn`t k2 7020

bjh`usj `atjr hneesfki ` tnrjj-ea-`-cfkm, tnjrj `twjlevj cfkms ljat areg wnfhn te hneesj twe-ea-`-cfkm. fs

]nj preb`bflfty ea ` aull neusj fs

T Saull neusjW 7k0k2k8k=

:2:

7 8, 9==2, :6>, 614

7 4.440=. (


]njrj `rj 2: 7 82 mfffljrjkt bfk`ry hemjs wftn : bfts. ]nj kugbjr ea hemwftn jx`htly 8 zjres jqu`ls tnj kugbjr ea w`ys ea hneesfki tnj bfts fk wnfhtnesj zjres ehhur. ]njrjaerj tnjrj `rj

:8

7 04 hemjs wftn jx`htly 8 zjres.

Trebljg 2.2.> Xelutfek

Xfkhj j`hn ljttjr h`k t`cj ek `ky ekj ea tnj = pessfblj ljttjrs fk tnj `lpn`bjtnj kugbjr ea 8 ljttjr werms tn`t h`k bj aergjm fs =8 7 1=. Fa wj `llew j`ljttjr te `ppj`r ekly ekhj tnjk wj n`vj = hnefhjs aer tnj ffrst ljttjr `kmhnefhjs aer tnj sjhekm `km twe hnefhjs aer tnj tnfrm ljttjr. ]njrjaerj, tnj`rj ` tet`l ea = 8 2 7 2= pessfblj hemjs.

:4


9/15

Gerjevjr, fa tnj ffrst shr`thnjm bex n`s tnj g`rc, tnjk tnjrj `rj =+ cg`rcbexjs eut eak0 rjg`fkfki bexjs. Hektfkufki tnfs `riugjkt, tnj preb`bflftn`t ` tfhcjt fs ` wfkkjr fs

p7: + c

k

= + c

k 0

8 + c

k 2

2 + c

k 8

0 + c

k =7

(c+ :)!(k :)!

c!k! . (

By h`rjaul hnefhj eak`km c, wj h`k hneesj p hlesj te 4.40. Aer jx`gplj,

k 6 00 0= 09c 4 0 2 8p 4.4496 4.402 4.404: 4.4464

(

@ i`gjh`rm wftn K7 0= bexjs `km : + c7 9 sn`mjm bexjs weulm bj qufrj`sek`blj.


(`) Xfkhj tnj preb`bflfty ea ` zjre fs 4.>, wj h`k jxprjss tnj preb`bflfty tnj hemj werm 44000 `s 2 ehhurrjkhjs ea ` 4 `km tnrjj ehhurrjkhjs ea0. ]njrjaerj

T S44000W 7 (4.>)2(4.2)8 7 4.44:02. (

(b) ]nj preb`bflfty tn`t ` hemj werm n`s jx`htly tnrjj 0s fs

T Stnrjj 0sW 7

:

8

(4.>)2(4.2)8 7 4.4:02. (

Trebljg 2.8.2 XelutfekIfvjk tn`t tnj preb`bflfty tn`t tnj Hjltfhs wfk ` sfkilj hn`gpfeksnfp fk `kifvjk yj`r fs 4.82, wj h`k ffkm tnj preb`bflfty tn`t tnjy wfk > str`fint KBhn`gpfeksnfps.

T S> str`fint hn`gpfeksnfpsW 7 (4.82)> 7 4.44400. (

::


10/15

]nj preb`bflfty tn`t tnjy wfk 04 tftljs fk 00 yj`rs fs

T S04 tftljs fk 00 yj`rsW 7

00

04

(.82)04(.1>) 7 4.444>=. (

]nj preb`bflfty ea j`hn ea tnjsj jvjkts fs ljss tn`k 0 fk 0444! Ifvjk tntnjsj jvjkts teec pl`hj fk tnj rjl`tfvjly snert ffaty yj`r nfstery ea tnj KB@

ft sneulm sjjg tn`t tnjsj preb`bflftfjs sneulm bj guhn nfinjr. Yn`t tgemjl evjrleecs fs tn`t tnj sjqujkhj ea 04 tftljs fk 00 yj`rs st`rtjm wnjk BRussjll oefkjm tnj Hjltfhs. Fk tnj yj`rs wftn Russjll (`km ` streki suppertfkh`st) tnj preb`bflfty ea ` hn`gpfeksnfp w`s guhn nfinjr.


Yj ckew tn`t tnj preb`bflfty ea ` irjjk `km rjm lfint fs 9/01, `km tn`t

` yjllew lfint fs 0/>. Xfkhj tnjrj `rj `lw`ys : lfints,I, V, `km R ebjy tgultfkegf`l preb`bflfty l`w?

T SI7 2, V 7 0, R7 2W 7 :!

2!0!2!

9

01

20

>

9

01

2. (

]nj preb`bflfty tn`t tnj kugbjr ea irjjk lfints jqu`ls tnj kugbjr ea rjlfints

T SI7 RW 7 T SI7 0, R7 0, V 7 8W + T SI7 2, R7 2, V 7 0W+ T SI7 4, R7 4, V 7 :W

7 :!

0!0!8!

9

01

9

01

0

>

8+

:!

2!0!2!

9

01

29

01

20

>

+ :!

4!4!:!

0

>

:

4.0==6. (

Trebljg 2.8.= Xelutfek

Aer tnj tj`g wftn tnj negjheurt `mv`kt`ij, ljt Yf `km Lf mjketj wnjtni`gjf w`s ` wfk er ` less. Bjh`usj i`gjs 0 `km 8 `rj negj i`gjs `km i`g2 fs `k `w`y i`gj, tnj trjj fs

:1


11/15

Y0p

L00p

Y20p

L2p

Y20p

L2p

Y8p

L80p

Y8p

L80p

Y0Y2 p(0p)

L0L2 p(0p)

Y0L2L8 p2(0p)

Y0L2Y8 p8

L0Y2Y8 p(0p)2

L0Y2L8 (0p)8

]nj preb`bflfty tn`t tnj tj`g wftn tnj negj heurt `mv`kt`ij wfks fs

T SNW 7 T SY0Y2W + T SY0L2Y8W + T SL0Y2Y8W

7p(0 p) +p8 +p(0 p)2. (

Ketj tn`t TSNW p aer 0/2 p 0. Xfkhj tnj tj`g wftn tnj negj heu`mv`kt`ij weulm wfk ` 0 i`gj pl`yeffl wftn preb`bflfty p, tnj negj heutj`g fs ljss lfcjly te wfk ` tnrjj i`gj sjrfjs tn`k ` 0 i`gj pl`yeffl!


(`) ]njrj `rj 8 ireup 0 cfhcjrs `km 1 ireup 2 cfhcjrs. Zsfki If te mjketn`t ` ireup f cfhcjr w`s hnesjk, wj n`vj

T SI0W 7 0/8, T SI2W 7 2/8. (

Fk `mmftfek, tnj prebljg st`tjgjkt tjlls us tn`t

T SC|I0W 7 0/2, T SC|I2W 7 0/8. (

Hegbfkfki tnjsj a`hts usfki tnj L`w ea ]et`l Treb`bflfty yfjlms

T SCW 7 T SC|I0W T SI0W + T SC|I2W T SI2W

7 (0/2)(0/8) + (0/8)(2/8) 7 9/0>. (

:9


12/15

Trebljg 2.=.0 Xelutfek

Areg tnj prebljg st`tjgjkt, wj h`k hekhlumj tn`t tnj mjvfhj hegpekjk`rj hekffiurjm fk tnj aellewfki w`y.

Y0

Y2

Y:

Y8

Y=

Y1

]e ffkm tnj preb`bflfty tn`t tnj mjvfhj wercs, wj rjpl`hj sjrfjs mjvfhjs 0, `km 8, `km p`r`lljl mjvfhjs : `km 1 j`hn wftn ` sfkilj mjvfhj l`bjljm wftn tnpreb`bflfty tn`t ft wercs. Fk p`rtfhul`r,

T SY0Y2Y8W 7 (0 q)8, (

T SY: Y1W 7 0 T SYh:Y

h1 W 7 0 q

2. (

]nfs yfjlms ` hegpesftj mjvfhj ea tnj aerg

0-q2

0-q

( )0-q 8

]nj preb`bflfty TSYW tn`t tnj twe mjvfhjs fk p`r`lljl werc fs 0 gfkus tnpreb`bflfty tn`t kjftnjr wercs?

T SYW 7 0 q(0 (0 q)8). (

Afk`lly, aer tnj mjvfhj te werc, betn hegpesftj mjvfhj fk sjrfjs gust wer]nus, tnj preb`bflfty tnj mjvfhj wercs fs

T SYW 7 S0 q(0 (0 q)8)WS0 q2W. (

14


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T SH7h, M7m, J7mW 7

=4!h!m!j!

hmj h + m + j7 =43 h,m,j 4,

4 etnjrwfsj.(

Trebljg 2.=.= Xelutfek

Areg tnj st`tjgjkt ea Trebljg 2.=.0, tnj hekffiur`tfek ea mjvfhj hegpekjkfs

Y0

Y2

Y:

Y8

Y=

Y1

By syggjtry, ketj tn`t tnj rjlf`bflfty ea tnj systjg fs tnj s`gj wnjtnjr wrjpl`hj hegpekjkt 0, hegpekjkt 2, er hegpekjkt 8. Xfgfl`rly, tnj rjlf`bflffs tnj s`gj wnjtnjr wj rjpl`hj hegpekjkt : er hegpekjkt 1. ]nus wheksfmjr tnj aellewfki h`sjs?

F Rjpl`hj hegpekjkt 0 Fk tnfs h`sj

T SY0Y2Y8W 7 (0 q

2)(0 q)2, (

T SY=W 7 0 q, (

T SY: Y1W 7 0 q2

. (]nfs fgplfjs

T SY0Y2Y8 Y=W 7 0 (0 T SY0Y2Y8W)(0 T SY=W)

7 0 q2

2(: =q+ q2). (

12


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Fk tnfs h`sj, tnj preb`bflfty tnj systjg wercs fs

T SYFW 7 T SY0Y2Y8 Y=W T SY: Y1W

7

0

q2

2(: =q+ q2)

(0 q2). (

FF Rjpl`hj hegpekjkt = Fk tnfs h`sj,

T SY0Y2Y8W 7 (0 q)8, (

T SY=W 7 0 q

2, (

T SY: Y1W 7 0 q2. (

]nfs fgplfjs


7 0 q

2+

q

2(0 q)8. (


T SYFFW 7 T SY0Y2Y8 Y=W T SY: Y1W

7

0 q2

+ q2

(0 q)8

(0 q2). (0

FFF Rjpl`hj hegpekjkt : Fk tnfs h`sj,

T SY0Y2Y8W 7 (0 q)8, (0

T SY=W 7 0 q, (0

T SY: Y1W 7 0 q2

2. (0

]nfs fgplfjs


7 (0 q)

0 + q(0 q)2

. (0

18


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T SYFFFW 7 T SY0Y2Y8 Y=W T SY: Y1W

7 (0 q)

0

q2

2

0 + q(0 q)2

. (0

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0 q

2+

q

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Trebljg 2.:.0 Xelutfek

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tnfs t`sc fs

aukhtfek SH,NW7twehefk(k)3

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T70-(H/=)3

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Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 2 Solutions