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SCHAUM'S
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Problem
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ProbabiRlaintdyo,Vm a riables,
andR andoPmr ocesses
· SeconEdd ition
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405 fulsloyl vperdo blems
Conceixspel aonafalc tloi uocrnossne c epts
•
Inforomnba itviaroranin advtoaemr i jaobdilinestst ,r ibution
•
funcmtoimoegnnestn, e fruantciPtnoigio psnrssoo,cn e sses,
Wiepnreorc peoswsseeprse d,ce tnrsaailntw dih einsto,ei se
USE WITHT HESEC OURSES
Probayb• i RlaintdoPmr oce•s Ssteosc haPsrtoicce• s Psreosb ayba inldi t
RandomV ariabIlnetsr odutcoPt rioobna bainldSi ttayt istics
•
HweHis uP,h .D.
SCHAUM'S
outlines
ProbabilRiatnyd,o mV ariables,
andR andomP rocesses
SCHAUM'S
outlines
ProbabilRiatnyd,o mV ariables,
andR andomP rocesses
--SecoEnddi tion -------------
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PROBABILIRTY,A NDOMV ARIABLEASN,D R ANDOMP ROCESSES
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Thipsu blicaitsdi eosni gnteopd r oviadcec uraatneda uthoritiantfiovrmea tiinro eng atrdo t hes ubjemcta ttecro veredl.it s s old
witthh eu nderstandtihnangte ithtehrea uthnoorr t hep ublisihsee nrg ageidnr enderlienggaa lc,c ountsiencgu,r ittriaedsi ng,
oro theprr ofessisoenrvailc eIsfl. e gaadlv icoero theerx pearts sistainsrc eeq uirtehdes, e rvicoefsa competepnrotf essional
persons houlbdes ought.
-From aDe claratioofnP rincipJloeisn tAldyo ptebdya Committoefte h e
AmericaBna rA ssociaatnidoa n C ommittoefeP ublishearsn dA ssociations
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1.P robabilities-sOyultllaeibtnic2e,..s R ,a ndomv ariables-Outslyilnleaesbt,ic 3,..S tochasptriocc esses-Outlines,
syllaebticI,..H suH,w eiP. (HwePii ao1)9,3 0S-chaumo'us tlionfte h eoaryn dp robleomfps r obabirlaintdyo,vm a riables,
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PrefatcoeT heS econEdd ition
Thep urposoef t hibso okl,i kiet psr evioeudsi tiiostn o,p roviadnei ntrodutcott ihoepn r inciopflp erso bability,
randovmari ableasn,dr andopmr ocesasnedths e iarp plications.
e
Th booik sd esignfeosdrt udeinntv sa rioduiss ciploifen negsi neersicnige,nm caet,h emataincdms a,n age
mentT.h eb ackgroruenqdu irteosd t udtyh eb ooki so ney earo fc alculeulse,m entdariyff erenetqiuaalt ions,
matrixan alysiansd, somes ignaanlds ystem theory, Fionurciletrurand sifnogrm sT.h eb ookc anbe useasd as elf
contaitneexdt booorfk o sre lf-sEtaucdhyt .o piicsi ntroducienad c haptweirt nhu merosuosl vperodb lemTsh.e
solvperdo blecmosn stiatnui tnet egpraartol f th e text.
Thinse we ditiionnc luadnedse xpanthdesc onteonftt sh fier setd itiIonan d.d ittioor ne finemetnhtr outghhe
texttw,o n ews ectioonnsp robability-gfuenncetriaoantndis nm gart ingalheasv bee eand deandd an ewc hapter
oni nformatthieoonr hya sb eeand ded.
Iw istho th ank my granddaugEhltyesArin an Kre bs fohre lpimnegi nth ep reparatoifto hni rse visiIao lns.o
wistho e xpresmsy appreciattothi eoe nd itosritaaloff f t hMec Graw-HSiclhla umS'esri efso trh eciarre c,o oper
atioann,da ttentdieovno tteodt hepre paratioofthn e book.
HwEIP.Hsu
ShannondaetVl all lFeyo rgAeu,d uboPne,n nsylvania
PrefatcoeT heF irEsdti tion
Thep urpoosfet hibso oki st op roviadnei ntroducttoti hoepn r inciopflp erso babirlaintdyo,vm a riabalneds ,
randopmr ocesasnedst heaiprp lications.
Theb ooki sd esignfeodsr t udeinntv sa rioduiss ciploifen negsi neersicnige,nm caet,h emataincdms a,n age
mentI.tm ayb eu seda sa textboaonkd /oar s upplemteona tl clu rrecnotm parabtleex tIsts. h ouladl sboe
useftuolt hosien tereisntt ehdefi elodf s elf-stTuhdeby o.o kc ombintehsea dvantaogfbe ost thh et extboaonkd
thes o-calrleevdi ebwo okI.tp rovidtehset extueaxlp lanatoifto hnets e xtboaonkdi, n t hed irewcaty c,h aracter
istoifct hree viebwo oki,tg ivehsu ndreodfcs o mpletely psroobllveemtdsh auts ee ssenthteioarlany d techniques.
Moreovetrh,es o lvperdo blearmesa ni ntegpraarlot ft het exTth.e b ackgroruenqdu itroes dt udtyh beo oki so ne
yeaorf c alculeulse,m entdairffye renetqiuaalt iomnastr,i xa nalysainsds, o mes ignaanld s ystetmh eoriyn,c lud
ingF ouriterra nsforms.
Iw istho t hanDkr .G ordoSni lvermfaonhr i isn valuasbulgeg estainodnc sr itirceavli eowft hem anuscript.
Ia lswoi sht oe xpremsys a ppreciattoti hoeen d itorsitaaolfff t heM cGraw-HiSlclh auSme rifeosrt heciarr e,
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HWEIP.Hsu
MontvillNee,wJ ersey
Contents
CHAPTER1 Probability 1
1.1I ntroduction 1
1.2S amplSep acaen dE vents 1
1.3A lgeborfaS ets 2
1.4P robabiSlpiatcye 6
1.5E qualLliyk elEyv ents 9
1.6C onditiPornoabla bility 10
1.7T otaPlr obability 10
1.8I ndependEevnetn ts 11
SolvePrdo blems 12
CHAPTER2 RandoVma riables so
2.1I ntroduction 50
2.2R andomV ariables 50
2.3D istribuFtuinocnt ions 52
2.4D iscreRtaned om VariabalnedPs r obabiMlaistsFy u nctions 53
2.5C ontinuRoanudso m VariablaensdP robabiDleintsyi Ftuyn ctions 54
2.6M eana ndV ariance 54
2.7S omeS peciDails tributions 55
2.8C onditiDoinsatrli butions 64
SolvePrdo blems 65
CHAPTER3 MultiRpalned oVma riables 101
3.1I ntroduction 101
3.2B ivariRaatned omV ariables 101
3.3J oinDti stributFiuonnc tions 102
3.4D iscrReatned omV ariables-JProoibnatb ilMiatsysF unctions 103
3.5C ontinuRoaunsd omV ariables-PJrooibnatb iDleintsyi Ftuyn ctions 104
3.6C onditiDoinsatrli butions 105
3.7C ovarianacned C orrelatCiooenffi cient 106
3.8C onditiMoenaanls a ndC onditiVoanraila nces 107
3.9N -VariaRtaen domV ariables 108
3.1S0p eciDails tributions 110
SolvePdr oblems 112
Contents
CHAPTER4 FunctioofnR sa ndoVma riabElxepse,c tatLiiomniT,th eorems 149
4.1 149
Introduction
4.2F unctioofnO sn eR andomV ariable 149
4.3 150
FunctioofnTw so RandomV ariables
4.4 151
Functioofnn s R andomV ariables
4.5 152
Expectation
>
4.6 154
Probability-GenFeurnacttiinogn s
4.7 155
Moment-GenerFautnicntgi ons
4.8 156
CharacterFiusntcitci ons
4.9 158
TheL awso fL argNeu mberasn dt heC entrLailm iTth eorem
159
SolvePdr oblems
CHAPTER5 RandomP rocesses 207
5. 1 207
Introduction
5.2 207
RandomP rocesses
5. 3 208
CharacteriozfaR tainodno mP rocesses
5. 4 209
ClassificaoftR iaonnd omP rocesses
5.5 211
Discrete-ParMaamrektoCevhr a ins
5. 6 216
PoissPorno cesses
5.7 218
WienePrr ocesses
5. 8 219
Martingales
222
SolvePdr oblems
CHAPTER6 AnalysainsdP rocessoifnR ga ndomP rocesses 271
6.1 271
Introduction
6.2C ontinuDiitffye,r enti,Ia nttieognration 271
6.3P oweSrp ectDreanls ities 273
6.4W hitNeo ise 275
6.5R esponosfeL ineaSry stetmosR andomI nputs 276
6.6F ouriSeerr iaensd K arhunen-LoEexvpea nsions 279
6.7F ouriTerra nsfoorfRm a ndomP rocesses 280
SolvePdr oblems 282
CHAPTER7 EstimatTihoeno ry 312
7.1I ntroduction 312
7. 2 ParametEesrt imation 312
7. 3 PropertoifPe osi nEts timators 312
7.4M aximum-LikelEishtoiomda tion 313
7.5B ayesE's timation 314
7.6M eanS quarEes timation 314
7.7L ineMaera nS quarEes timation 315
SolvePdr oblems 316
CHAPTER8 DecisiTohne ory 331
8. 1 Introduction 331
8.2H ypotheTseisst ing 331
8. 3 DecisiToens ts 332
SolvePdr oblems 335
Contents
CHAPTER9 QueueiTnhge ory 349
9.1 Introduction 349
9.2 QueueiSnygs tems 349
9.3 Birth-DePartohc ess 350
9.4 TheM /M/1Q ueueiSnygs tem 352
9.5 TheM IM/sQ ueueiSnygs tem 352
9.6 The M/M/1Q/uKe ueiSnygs tem 353
9.7 The M/M/s/KQueuSeyisntge m 354
SolvePdr oblems 355
CHAPTER1 0 InformatTihoenor y 367
10.1I ntroduction 367
10.2M easuroefI nformation 367
10.3D iscrMeetmeo ryleCshsa nnels 369
10.4M utuaIln formation 371
10.5C hannCealp acity 373
10.6C ontinuCohuasn nel 374
10.7A dditiWvhei tGea ussiNaoni sCeh annel 375
10.8S ourcCeo ding 376
10.9E ntroCpoyd ing 378
SolvePdr oblems 380
APPENDIAX NormaDli stribution 411
APPENDIBX FouriTerra nsform 413
B.l ContinuToiumse-F ouriTerra nsform 413
B.2 DiscreTtiem-eF ouriTerra nsform 414
INDEX 417
Probability
1.1 Introduction
Thes tudyo fp robabisltietmfryso mth ean alysoifsc ertgaaimne osf c hancaen,di th asf ounadp plicaitnim oonsst
branchoefss c ienacnede ngineerIintn hgi.cs h apttehreb asicco ncepotfps r obabitlhietoyary r ep resented.
1.2 SamplSep acea ndE vents
A. RandoEmx periments:
Int hes tudoyfp robabialniypt ryo,c eosfos b servaitsrei fone rretdo a sa ne xperiTmheern ets.uol fta sn o bser
vatiaornec alltehdeo utcoomfte hsee xperimAenn etx.p erimiescn atl laera dn doemx periimfi etosnu tt come
cannobtep redicTtyepdi.c eaxla mploefsa r andoemx perimaernett h er ololfa d iet,h et ososfa c oind,r awing
ac ardfr oma d ecko,rs electaim negs sasgieg nfaoltrr a nsmissfiroonsm e vermaels sages.
B. SamplSep ace:
The seotf a lplo ssiobultec omoefsa r andoemx perimiescn atl ltehdse ampslpea (coeur nivesrestaa)nld,i t i s
denotbeydS .A n elemeinntS isc allaes da mpploei Enatc.ho utcomoefa r andoemx perimceonrrte spotnod s
as amplpeo int.
EXAMPLE 1.1 Fintdh es amplsep acfeo trh eex perimoeftnots sianc go i(na o)n caen d( bt)w ice.
(a)Th eraree t wop ossiobultec omheesa,d osrt aiTlhsu.s :
S ={ H,T }
wherHean d Trepresheenatd antda irle,s pectively.
(b)Th eraree f oupro ssiobultec omTehse.y a rep aiorfsh eadasn dt aiTlhsu.s :
S ={ HH,H T,T H,T T}
EXAMPLE 1.2 Finthde samplsep acfeo trh eex perimeonftt o ssianc go irenpe atedlaynd o fc ountithneg n um
ber oft ossreeqsu iruendt tihlfie r shte aadp pears.
Clearlayl plo ssiobultec omfeostr h iesx perimareent th tee rmosft hes equen1c,2e ,3 ,. .T.h us:
S={l,2,3.,.} .
Notteh atth eraer ea ni nfinintuem beorf o utcomes.
