7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Puoz 0.0 Whdutohk

Ok tmb ^bkk lo`cr`is nhr p`rts (`)-(c) ebdhw, tmb sm`lbl `rb` rbprbsbkts tmoklog`tbl sbt.

I H

\

I H

\

(`) K9\g (e) KI

I H

\

I H

\

(g) KI (l) \g Ig

Puoz 0.5 Whdutohk

@09{vvv,vvl,lvv,lvl}

E09{vlv, vll, llv, lll}@59{vvv,lll}

E59{vlv, lvl}

@79{vvv, vvl, vlv, lvv,vll, lvl, llv}

E79{lll, llv, lvl,vll}

Ubg`dd tm`t@o `klEo `rb ghddbgtovbdy bxm`ustovb on@oEo 9W. @dsh, @o `k

Eo `rb iutu`ddy bxgdusovb on@o Eo 9 . Wokgb wb m`vb wrottbk lhwk b`p`or @o `kl Eo `ehvb, wb g`k soipdy gmbgj nhr tmbsb prhpbrtobs.

\mb p`or@0`klE0`rb iutu`ddy bxgdusovb `kl ghddbgtovbdy bxm`ustovb. \p`or @5 `kl E5 `rb iutu`ddy bxgdusovb eut kht ghddbgtovbdy bxm`ustovb. \p`or @7 `kl E7 `rb kht iutu`ddy bxgdusovb sokgb lvl ebdhkcs th @7 `kl EMhwbvbr,@7 `kl E7 `rb ghddbgtovbdy bxm`ustovb.

5

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Qrhedbi 0.0.0 Whdutohk

E`sbl hk tmb ^bkk lo`cr`i hk tmb rocmt, tmb ghipdbtb Cbr-d`kl`s pozz` ibku os

Ubcud`r wotmhut thppokcs Ubcud`r wotm iusmrhhis

Ubcud`r wotm hkohks Ubcud`r wotm iusmrhhis `kl hkohks \usg`k wotmhut thppokcs \usg`k wotm iusmrhhis

I

\

Qrhedbi 0.0.5 Whdutohk

E`sbl hk tmb bkk lo`cr`i hk tmb rocmt, tmb `kswbrs `rbihstdy n`ordy str`ocmtnhrw`rl. \mb hkdy trogjokbss os tm`t `pozz` os botmbr \usg`k (\) hr Kb`phdot`k (K) sh {K, \} os `p`rtotohk eut tmby `rb kht lbpogtbl `s ` p`rtotohk. Wpbgoffg`ddy,tmb bvbkt Kos tmb rbcohk hn tmb ^bkk lo`cr`i hutsolb hn tmbsqu`rb edhgj hn bvbkt \. On tmos os gdb`r, tmb qubstohks `rbb`sy.

I

\

(`) WokgbK9\g,KI9. \musK `klI`rb kht iutu`ddy bxgdusov

(e) Bvbry pozz` os botmbr Kb`phdot`k (K), hr \usg`k (\). MbkgbK\ 9sh tm`t K `kl \ `rb ghddbgtovbdy bxm`ustovb. \mus ots `dsh (trovo`ddtrub tm`t K \ I 9 W. \m`t os,U, \ `kl I`rb `dsh ghddbgtovbbxm`ustovb.

(g) Nrhi tmb bkk lo`cr`i,\`klH `rb iutu`ddy bxgdusovb. Ok whrls, tmib`ks tm`t \usg`k pozz`s kbvbr m`vb hkohks hr pozz`s wotm hkohks `kbvbr \usg`k. @s `k `solb, \usg`k os ` n`jb pozz` lbsock`tohk: hsmhudlkt ghkgdulb tm`t pbhpdb nrhi \usg`ky `gtu`ddy losdojb hkohks.

(l) Nrhi tmb bkk lo`cr`i, I \ `kl H `rb iutu`ddy bxgdusovb. \mCbrd`kl`s lhbskt i`jb \usg`k pozz` wotm iusmrhhis `kl hkohks.

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7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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(b) \mb bvbkt tm`t ihrb tm`k hkb gorguot os `ggbpt`edb os

G9 {```,``n,`n`,n``} . (

\mb bvbkt tm`t `t db`st twh gorguots n`od os

L9 {nn`,n`n,`nn,nnn} . (

(n) Okspbgtohk smhws tm`t G L9 sh G `kl L `rb iutu`ddy bxgdusovb

(c) Wokgb G L9W, G`kl L `rb ghddbgtovbdy bxm`ustovb.

Qrhedbi 0.5.7 Whdutohk

\mb s`ipdb sp`gb os

W9 {@, . . . , J , @, . . . , J , @, . . . , J , @, . . . , J } . (

\mb bvbkt M os tmb sbt

M9 {@, . . . , J } . (

Qrhedbi 0.5.< Whdutohk

\mb s`ipdb sp`gb os

W9

0/0 . . . 0/70, 5/0 . . . 5/5?, 7/0 . . . 7/70, /71, 3/0 . . . 3/70, =/0 . . . =/70,?/0 . . . ?/70, 01/0 . . . 01/70, 00/0 . . . 00/71, 05/0 . . . 05/70

. (

\mb bvbkt M lbffkbl ey tmb bvbkt hn ` Fudy eortml`y os lbsgroebl ey nhdhwokc 70 s`ipdb phokts.

M9 {3/0, 3/5, . . . , 3/70} . (

>

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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7. On wb kbbl th gmbgj wmbtmbr b`gm rbsost`kgb lhbskt n`dd ebdhw ` iokiui v`dub (ok tmos g`sb 21 hmis nhr U0`kl 011 hmis nhr U5), ` p`rtotohos G0, G5, G7, G< wmbrb

G09 {U0 421, U54011} , G59 {U0421, U5 011} , (

G79 {U0 21, U54011} , G

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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(`) \mus

QZU7S5Y 9 QZU7Y QZS5Y 9 0

7>.

(e) Ok n`gt, b`gm p`or hn phssoedb rhdds UoSf m`s prhe`eodoty 0/7>. \h ffkQZW2Y, wb `ll up tmb prhe`eodotobs hn `dd p`ors tm`t sui th 26

QZW2Y 9 QZU0S, >)} . (

wotm 7> hutghibs, b`gm wotm prhe`eodoty 0/7> Khtb tm`t tmb bvbkt tm`t tm`eshdutb v`dub hn tmb lofibrbkgb hn tmb twh rhdds bqu`ds 7 os

L79 {(0, ), (, 7)} . (

Wokgb tmbrb `rb > hutghibs ok L7, QZL7Y 9 >/7> 9 0/>.

Qrhedbi 0.7.< Whdutohk

(`) N@DWB. Wokgb QZ@Y 9 0 QZ@gY 9 5 QZ@gY oipdobs QZ@gY 9 0/7.

(e) N@DWB. Wupphsb@9 E `kl QZ@Y 9 0/5. Ok tm`t g`sb,

Q Z@EY 9 Q Z@Y 9 0/5; 0/< 9 Q Z@Y Q ZEY . (

(g) \URB. Wokgb@E @, QZ@EY QZ@Y, \mos oipdobs

Q Z@EY Q Z@Y4 Q ZEY . (

(l) N@DWB6 Nhr ` ghuktbrbx`ipdb, dbt @ 9 `kl QZEY ; 1 sh tm`t @ @ E9 `kl QZ@Y 9 QZ@ EY 9 1 eut 1 9 QZ@Y4 QZEY.

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7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Qrhedbi 0.7.= Whdutohk

Dbt so lbkhtb tmb hutghib tm`t tmb lhwk n`gb m`s o lhts. \mb s`ipdb sp`os W9 {s0, . . . , s>}. \mb prhe`eodoty hn b`gm s`ipdb hutghib os QZsoY 9 0/Nrhi \mbhrbi 0.0, tmb prhe`eodoty hn tmb bvbkt Btm`t tmb rhdd os bvbk os

Q ZBY 9 Q Zs5Y + Q ZsY 9 7/>. (

Qrhedbi 0.7.? Whdutohk

Dbt so bqu`d tmb hutghib hn tmb stulbkts quoz. \mb s`ipdb sp`gb os tmghiphsbl hn `dd tmb phssoedb cr`lbs tm`t smb g`k rbgbovb.

W9 {1, 0, 5, 7, , 3, =, ?, 01} . (

Wokgb b`gm hn tmb 00 phssoedb hutghibs os bqu`ddy dojbdy, tmb prhe`eodoty rbgbovokc ` cr`lb hno, nhr b`gmo 9 1, 0, . . . , 01 os QZsoY 9 0/00. \mb prhe`eodotm`t tmb stulbkt cbts `k @ os tmb prhe`eodoty tm`t smb cbts ` sghrb hn ? hmocmbr. \m`t os

Q ZCr`lb hn @Y 9 Q Z?Y + Q Z01Y 9 0/00 + 0/00 9 5/00. (

\mb prhe`eodoty hn n`odokc rbquorbs tmb stulbkt th cbt ` cr`lb dbss tm`k

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Khtb tm`t sh n`r wb m`vb usbl hkdy @xohi 7. Nok`ddy, wb hesbrvb tm@ E g`k eb wrottbk `s tmb ukohk hn iutu`ddy bxgdusovb bvbkts

@ E 9 (@E) (@Eg) (@gE). (

Hkgb `c`ok, usokc \mbhrbi 0.7, wb m`vb

QZ@ EY 9 QZ@EY + QZ@EgY + QZ@gEY (0

Wuestotutokc tmb rbsudts hn Bqu`tohk (=) okth Bqu`tohk (01) yobdls

Q Z@ EY 9 Q Z@EY + Q Z@Y Q Z@EY + Q ZEY Q Z@EY , (0

wmogm ghipdbtbs tmb prhhn. Khtb tm`t tmos gd`oi rbquorbl hkdy @xohi

(l) Hesbrvb tm`t sokgb @ E, wb g`k wrotb E `s tmb iutu`ddy bxgdusoukohk E 9@ (@gE). Ey \mbhrbi 0.7 (wmogm usbs @xohi 7),

Q ZEY 9 Q Z@Y + Q Z@gEY . (0

Ey @xohi 0, QZ@gEY 1, mogm oipdobs QZ@Y QZEY. \mos prhhn us@xohis 0 `kl 7.

Qrhedbi 0.. (

\mb prhe`eodoty ` erobn g`dd wodd m`vb kh m`klhfis os

Q ZM1|EY 9Q ZM1EY

Q ZEY 9

1.9

5

7. (

03

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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(e) \mb prhe`eodoty hn hkb m`klhfi os QZM0Y 9 QZM0EY + Q ZM0DY 9 1.5. \mprhe`eodoty tm`t ` g`dd wotm hkb m`klhfi wodd eb dhkc os

Q ZD|M0Y 9Q ZM0DY

Q ZM0Y 9

1.0

1.59

0

5. (

(g) \mb prhe`eodoty ` g`dd os dhkc os QZDY 9 0QZEY 9 1.}.

(`) Wokgb C0 9 {s5, s7, s} `kl `dd hutghibs m`vb prhe`eodoty 0/QZC0Y 9 2/>. \mb bvbkt U7C09 {s7} `kl QZU7C0Y 9 0/> sh tm`t

Q ZU7|C0Y 9Q ZU7C0Y

Q ZC0Y 9

0

2. (

(e) \mb ghklotohk`d prhe`eodoty tm`t > os rhddbl covbk tm`t tmb rhdd os crb`ttm`k 7 os

Q ZU>|C7Y 9Q ZU>C7Y

Q ZC7Y 9 Q Zs>Y

Q ZsY 90/>

7/> . (

(g) \mb bvbktBtm`t tmb rhdd os bvbk os B9 {s5, s} `kl m`s prhe`eodo7/>. \mb fhokt prhe`eodoty hnC7 `kl B os

Q ZC7BY 9 Q ZsY 9 0/7. (

0=

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Qrhedbi 0.

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

11/13

Qrhedbi 0.>.< Whdutohk

@

E

Ok tmb ^bkk lo`cr`i, `ssuib tmb s`ipdb sp`gm`s `rb` 0 ghrrbsphklokc th prhe`eodoty 0. @lr`wk, ehtm @ `kl E m`vb `rb` 0/< sh tm`QZ@Y 9 QZEY 9 0/ `kl ghvbrs 0/< hn@ `kl 0/hnE. \m`t os, @`kl E `rb oklbpbklbkt sokgb

Q Z@EY 9 Q Z@Y Q ZEY . (0

Qrhedbi 0.>.2 Whdutohk

(`) Wokgb@ `kl E `rb iutu`ddy bxgdusovb, QZ@ EY 9 1. Wokgb QZ@ EY 1,

Q Z@ EY 9 Q Z@Y + Q ZEY Q Z@ EY 9 7/=. (

@ ^bkk lo`cr`i smhudl ghkvokgb yhu tm`t @ Eg sh tm`t@Eg 9@

\mos oipdobs

Q Z@ EgY 9 Q Z@Y 9 0/.> Whdutohk

(`) Wokgb G`kl L `rb oklbpbklbkt,

Q ZG LY 9 Q ZGY Q ZLY 9 02/>

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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Qrhedbi 0.>.? Whdutohk

Nhr ` s`ipdb sp`gbW9 {0, 5, 7, lostokgt bqu`ddy dojbdy hutghibs nhr tmb sbghkl cbkbr`tohk hn ppd`kts e`sbl hk ` ffrst cbkbr`tohk hn{rwyc,rwcy,wryc,wrcy}. \mbsb `rb

rryy rryc rrcy rrcc

rwyy rwyc rwcy rwccwryy wryc wrcy wrccwwyy wwyc wwcy wwcc

@ pd`kt m`s ybddhw sbbls, tm`t os bvbkt _hggurs, on ` pd`kt m`s `t db`st hklhiok`kt y cbkb. Bxgbpt nhr tmb nhur hutghibs wotm ` p`or hn rbgbssovbcbkbs, tmb rbi`okokc 05 hutghibs m`vb ybddhw sbbls. Nrhi tmb `ehvb, wb s

tm`t

Q Z_Y 9 05/0> 9 7/< (

`kl

Q ZUY 9 05/0> 9 7/

7/21/2019 Probability and Stochastic Processes 3rd Edition Roy D. Yates Chapter 1 Solutions

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\h ffkl tmb ghklotohk`d prhe`eodotobs QZU|_Y `kl QZ_|UY, wb ffrst iust ffkQZU_Y. Khtb tm`t U_, tmb bvbkt tm`t ` pd`kt m`s rhuklbl ybddhw sbbls, tmb sbt hn hutghibs

U_ 9 {rryy,rryc,rrcy,rwyy,rwyc,rwcy,wryy,wryc,wrcy} . (

Wokgb QZU_Y 9 ?/0>,

Q Z_ |U Y 9Q ZU_Y

Q ZUY 9

?/0>

7/< 9 7/< (

`kl

Q ZU |_Y 9Q ZU_Y

Q Z_Y 9

?/0>

7/< 9 7/ Q ZU_gY 9 7/0>, (

Q ZUg_Y 9 7/0> Q ZUg_gY 9 0/0>. (

Qrhedbi 0.>.00 Whdutohk

(`) Nhr `ky bvbkts @ `kl E, wb g`k wrotb tmb d`w hn tht`d prhe`eodoty tmb nhri hn

Q Z@Y 9 Q Z@EY + Q Z@EgY . (

Wokgb @`kl E `rb oklbpbklbkt, QZ@EY 9 QZ@Y QZEY. \mos oipdobs

Q Z@EgY 9 Q Z@Y Q Z@Y Q ZEY 9 Q Z@Y (0 Q ZEY) 9 Q Z@Y Q ZEgY . (

\mus@ `klEg `rb oklbpbklbkt.