南京大学随机过程练习题附中文解释及答案

南京⼤学随机过程练习题附中⽂解释及答案第⼆章RandomVariables随机变量1、（2.16）Anairlineknowsthat5percentofthepeoplemakingreservationsonacertainflightwillnotshowup.Consequently,theirpolicyistosell52ticketsforaflightthatcanholdonly50passengers.Whatistheprobabilitythattherewillbeaseatavailableforeverypassengerwhoshowsup?航空公司知道预订航班的⼈有5%最终不来搭乘航班。因此，他们的政策是对于⼀个能容纳50个旅客的航班售52张票。问每个出现的旅客都有位答：05.0*95.0*52-95.0-15152）()(2、(2.25略变动)Supposethattwoteamsareplayingaseriesofgames,eachofwhichisindependentlywonbyteamAwithprobabilitypandbyteamBwithprobability1-p.Thewinneroftheseriesisthefirstteamtowinigames.Ifi=4,findtheprobabilitythatatotalof7gamesareplayed.Findthepthatmaximizes/minimizesthisprobability.假定两个队玩⼀系列游戏，A队独⽴地赢的概率是p，B队独⽴地赢的概率是1-p。先赢i次游戏的队为胜利者。若i=4，求总共进⾏了7次游戏的概率。求出使这个概率最⼤/最⼩的p值。答：Atotalof7gameswillbeplayedifthefirst6resultin3winsand3losses.Thus,P{7games}=33)1(36pp-.Differentiationyieldsthat])1(3)1(3[20}7{2332ppppPdpd---==]21[)1(6022ppp--.Thus,thederivativeiszerowhenp=1/2.Takingthesecondderivativeshowsthatthemaximumisattainedatthisvalue.3、(2.27)Afaircoinisindependentlyflippedntimes,ktimesbyAandn-ktimesbyB.FindthattheprobabilitythatAandBflipthesamenumberofheads.⼀枚均匀的硬币独⽴地抛掷n次，k次由A抛掷，n-k次由B抛掷。证明A和B抛掷出相同次正⾯的概率等于总共有k次正⾯的概率。答：P{samenumberofheads}=∑==iiBiAP},{=∑-2/1()2/1()2/1(innnkniknikkiknikAnotherargumentisasfollows.P{#headsofA=#headsofB}=P{#tailsofA=#headsofB}sincecoinisfair=P{k?#headsofA=#headsofB}=P{k=total#heads}.4、(2.37)LetX1,X2,…,Xnbeindependentrandomvariables,eachhavingauniformdistributionover(0,1).LetM=maximum(X1,X2,…,Xn).FindthedistributionfunctionofM.令X1,X2,...,Xn是独⽴随机变量，每个都是（0,1）上的均匀分布。令M=max(X1,X2,...,Xn)。求解M的分布函数。5、(2.43)Anurncontainsn+mballs,ofwhichnareredandmareblack.Theyarewithdrawnfromtheurn,oneatatimeandwithoutreplacement.LetXbethenumberofredballsremovedbeforethefirstblackballischosen.WeareinterestedindeterminingE[X].⼀个瓮中含有n+m个球，其中n个红球，m个⿊球。它们⼀次⼀个从瓮中不放回地被抽取。以X记在⾸次取得⿊球前取出的红球个数。我们关⼼的是确定E[X]。（书上还有：为了得到这个量，将球⽤1到n的数字标记。现在随机变量Xi，i=1，ⅆ,n，定义为Xi=1，若红球i在任意⿊球前取出，Xi=0，其他情形。（a）⽤Xi表⽰X，（b）求E[X]6、(2.64)Showthatthesumofindependentidenticallydistributedexponentialrandomvariableshasagammadistribution.证明独⽴同分布的指数随机变量之和有伽马分布。7、(2.77)LetXandYbeindependentnormalrandomvariables,eachhavingparametersµand2σ.ShowthatX+YisindependentofX-Y.σ。证明X+Y与X-Y假设X和Y是独⽴正态随机变量，都具有均值µ和⽅差28、(3.8)Anunbiaseddieissuccessivelyrolled.LetXandYdenote,respectively,thenumberofrollsnecessarytoobtainasixandafive.Find(a)E[X],(b)E[X|Y=1]相继地掷⼀颗不均匀的骰⼦。令X和Y分别记得到⼀个6和⼀个5所必须的抛掷次数。求（a）E[X]，（b）E[X|Y=1]。E[X]=E[X|C1]P(C1)+E[X|C2]P(C2)+...+E[X|Cn]P(Cn)9、(3.23)Acoinhavingprobabilitypofcomingupheadsissuccessivelyflippeduntiltwoofthemostrecentthreeflipsareheads.LetNdenotethenumberofflips.(Notethatifthefirsttwoflipsareheads,thenN=2).FindE[N].连续地掷⼀枚出现正⾯的概率为p的硬币，直⾄最近的三次抛掷中有两次是正⾯。以N记炮制的次数（注意，如果前两次抛掷的结果都是正⾯，则N=2）。求E[N]。10、(3.26)Youhavetwoopponentswithwhomyoualternateplay.WheneveryouplayA,youwinwithprobabilitypA;wheneveryouplayB,youwinwithprobabilitypB,wherepB>pA.Ifyouobjectiveistominimizethenumberofgamesyouneedtoplaytowintwoinarow,shouldyoustartwithAorwithB?你有两个对⼿与你轮番博弈。与A博弈时你赢的概率是pA，⽽与B博弈时你赢的概率是pB，且pB>pA。如果你的⽬标是使你连赢两次所需的博弈次数最少，你应和A还是和B开始？1、(4.23)Trialsareperformedinsequence.Ifthelasttwotrialsweresuccesses,thenthenexttrialisasuccesswithprobability0.8;otherwisethenexttrialisasuccesswithprobability0.5.Inthelongrun,whatproportionoftrialsaresuccesses?试验依次地进⾏。如果最后的两次试验是成功，那么下⼀次试验以概率0.8是成功；否则下⼀次以概率0.5是成功。在长程中，成功的⽐例是多2、(4.24)Considerthreeurns,onecoloredred,onewhite,andoneblue.Theredurncontains1redand4blueballs;thewhiteurncontains3whiteballs,2redballs,and2blueballs;theblueurncontains4whiteballs,3redballs,and2blueballs.Attheinitialstage,aballisrandomlyselectedfromtheredurnandthenreturnedtothaturn.Ateverysubsequentstage,aballisrandomlyselectedfromtheurnwhosecoloristhesameasthatoftheballpreviouslyselectedandisthenreturnedtothaturn.Inthelongrun,whatproportionoftheselectedballsarered?Whatproportionarewhite?Whatproportionareblue?考察红、⽩、蓝三个坛⼦。红⾊的坛⼦含有1个红球，4个蓝球；⽩⾊的坛⼦含有3个⽩球，2个红球，2个蓝球；蓝⾊的坛⼦含有4个⽩球，3个红球，2个蓝球。开始时随机地从红⾊的坛⼦中任取⼀个球，然后放回这个坛⼦。在随后的每⼀步，从颜⾊与前⼀个取得的球相同的坛⼦中随机取出⼀个球，然后放回这个坛⼦。在长程中，取得红球的概率是多少？取得⽩球的概率是多少？取得蓝球的概率是多少？3、(4.32)Eachoftwoswitchesiseitheronoroffduringaday.Ondayn,eachswitchwillindependentlybeonwithprobability[1+#ofonswitchesduringdayn-1]/4.Forinstance,ifbothswitchesareonduringdayn-1,theneachwillindependentlybeonduringdaynwithprobability3/4.Whatfractionofdaysarebothswitcheson?Whatfractionsarebothoff?在⼀天中两个开关或者开或者关。在第n天，每个开关独⽴地处于开的概率是[1+第n-1天是开的开关数]/4。例如，如果在第n-1天两个开关都是开的，那么在第n天，每个开关独⽴地处于开的概率是3/4。问两个开关都是开的天数的⽐例是多少？两个开关都是关的天数的⽐例是多少？4、(4.41)Letridenotethelong-runproportionoftimeagivenirreducibleMarkovchainisinstatei.Explainwhyriisalsotheproportionoftransitionsthatareintostateiaswellasbeingtheproportionoftransitionthatarefromstatei.以ri记⼀个给定的不可约的马尔可夫链处在状态i的长程时间⽐例。解释为什么ri也是进⼊状态i的转移⽐例，同时是从状态i除了的转移⽐例。Thenumberoftransitionsintostateibytimen,thenumberoftransitionsoriginatingfromstateibytimen,andthenumberoftimeperiodsthechainisinstateibytimenalldifferbyatmost1.Thus,theirlongrunproportionsmustbeequa。l5、(4.44)Supposethatapopulationconsistsofafixednumber,say,m,ofgenesinanygeneration.Eachgeneisoneoftwopossiblegenetictypes.Ifanygenerationhasexactlyi(ofitsm)genesbeingtype1,thenthenextgenerationwillhavejtype1geneswithprobabilityjmjmimmijm--.LetXndenotethenumberoftype1genesinthenthgeneration,andassumethatX0=i.(a)FindE[Xn](b)Whatistheprobabilitythateventuallyallthegeneswillbetype1?假设⼀个总体在其任意⼀代由定值m个基因组成。每个基因是两个可能的基因型之⼀。如果任意⼀代（在它的m个中）恰有i个基因是1型，那么下⼀代以概率jmjmimmijm--有j个1型（和m-j个2型）基因。以Xn记第n代的1型基因个数，并假定X0=i。（a）求E[Xn]，（b）最终所有的基因都是1型的概率是多少？6、(4.47)Let{Xn,n>=0}denoteanergodicMarkovchainwithlimitingprobabilitiesri.Definetheprocess{Yn,n>=1}byYn={Xn-1，Xn}.Thatis,Ynkeepstrackofthelasttwostatesoftheoriginalchain.Is{Yn}aMarkovchain?Ifso,determineitstransitionprobabilitiesandfind)},({limjiYPnn=∞>-以{Xn,n>=0}记具有极限概率ri的⼀个遍历的马尔可夫链。以Yn={Xn-1，Xn}定义过程{Yn,n>=1}。即Yn追踪原来链的最后两个状态。问{Yn,n>=1}是否是⼀个马尔可夫链？如果是，确定它的转移概率，并求)},({limjiYPnn=∞>-。7、(4.54)Mballsaredistributedbetweentwourns,andateachtimepointoneoftheballsischosenatrandomandisthenremovedfromitsurnandplacedintheotherone.LetXndenotethenumberofmoleculesinurn1afterthenthswitchandletun=E[Xn].Findun+1asafunctionofun.M个球分布于两个坛⼦中，在每个时间点，随机地取出⼀个球，然后将它移出它的坛⼦，并放到另⼀个坛⼦中。以Xn记载第n次转移后在坛⼦1中的球数，并令un=E[Xn]。求un+1关于un的函数。8、(4.5.1例题)Consideragamblerwhoateachplayofthegamehasprobabilitypofwinningoneunitandprobabilityq=1-poflosingoneunit.Assumingthatsuccessiveplaysofthegameareindependent,whatistheprobabilitythat,startingwithiunits,thegambler’sfortunewillreachNbeforereaching0?考察⼀个赌徒，他在每次赌博中以概率p赢⼀个单位，并以概率q=1-p输⼀个单位。假设各次赌博都是独⽴的，赌徒在开始时有i个单位，问他的财富在达到0以前先达到N的概率是多少？答：如果我们以Xn记玩家在时间n的财富，那么{Xn,n>=0}是⼀个有转移概率P00=PNN=1，Pi,i+1=p=1-Pi,i-1，i=1,2,...,N-1的马尔可夫链。对初始的⼀次赌博的结果取条件，得到：Pi=pPi+1+qPi-1由p+q=1，将原式转化为pPi+qPi=pPi+1+qPi-1。化简，代⼊P0=0，可求得。再利⽤PN=1，得到P1和Pi。当N趋向于⽆穷时，因此，若p>0.5，则存在⼀个正概率，赌徒的财富将⽆限制地增长；⽽若p<=0.5，则赌徒将在对阵⼀个⽆限富有的对⼿时破产。9、(4.57)Aparticlemovesamongn+1verticesthataresituatedonacircleinthefollowingmanner.Ateachstepitmovesonestepeitherintheclockwisedirectionwithprobabilityporinthecounterclockwisedirectionwithprobabilityq=1-p.Startingataspecifiedstate,callitstate0,letTbethetimeofthefirstreturntostate0.FindtheprobabilitythatallstateshavebeenvisitedbytimeT.⼀个质点在位于圆周上的n+1个顶点间以如下⽅式移动：每次以概率p按顺时钟⽅向移动⼀步，或者以概率q=1-p按逆时钟⽅向移动⼀步。从⼀个特殊的状态0出发，令T是它⾸次回到状态0的时间。求在T以前⼀切状态都已访问遍的概率。1、(5.12)IfXi,i=1,2,3areindependentexponentialrandomvariableswithratesi,findP(X1如果Xi,i=1,2,3是速率为iλ（i=1,2,3）的独⽴指数随机变量，求P(X12、(5.20)Consideratwo-serversysteminwhichacustomerisservedfirstbyserver1,thenbyserver2,andthendeparts.Theservicetimesatserveriareexponentialrandomvariableswithratesiµ,i=1,2.Whenyouarrive,youfindserver1freeandtwocustomersatserver2——customerAinserviceandcustomerBwaitinginline.(a)FindPA,theprobabilitythatAisstillinservicewhenyoumoveovertoserver2.(b)FindPB,theprobabilitythatBisstillinthesystemwhenyoumoveovertoserver2.(c)FindE[T],whereTisthetimethatyouspendinthesystem.考虑有两条服务线的系统，顾客先接受服务线1服务，再到服务线2，然后离开。µ的指数随机变量，i=1,2。当你到达时，你发现服务线i的服务时间是速率为i服务线1有空，⽽在服务线2那⾥有两个顾客，顾客A在接受服务，顾客B在队中等候。（a）求当你到服务线2时，A还在接受服务的概率PA。（b）求当你到服务线2时，B还在接受服务的概率PB。（c）求E[T]，其中T是你在系统中的时间。3、（可能考）(5.24)Thereare2serversavailabletoprocessnjobs.Initially,eachserverbeginsworkonajob.Wheneveraservercompletesworkonajob,thatjobleavesthesystemandtheserverbeginsprocessinganewjob(providedtherearestilljobswaitingtobeprocessed).LetTdenotethetimeuntilalljobshavebeenprocessed.Ifthetimethatittakesserveritoprocessajobisexponentiallydistributedwithrateµ,findE[T]andvar(T).i有两条服务线处理n件零活。最初，每条服务线先处理⼀件零活。只要⼀条服务线完成了⼀件零活，这件零活就离开系统，并且这个服务线开始处理新的零活（当仍旧有等待处理的零活时）。以T记直到所有的零活都处理完的时间。如果服务线i处理⼀件零活的时间以速率iµ（i=1,2）指数的分布。求E[T]和var(T)。4、(5.30)ThelifetimesofA’sdogandcatareindependentexponentialrandomvariableswithrespectiveratesdλandcλ.Oneofthemhasjustdied.Findtheexpectedadditionallifetimeoftheotherpet.某⼈养的狗和猫的寿命分别是有速率dλ和cλ的独⽴的指数随机变量。其中⼀只刚刚死去。求另⼀只宠物的后续寿命。5、(5.31)Adoctorhasscheduledtwoappointments,oneat1pm,andtheotherat1:30pm.Theamountsoftimethatappointmentslastareindependentexponentialrandomvariableswithmean30minutes.Assumingthatbothpatientsareontime,findtheexpectedamountoftimethatthe1:30appointmentspendsatthedoctor’soffice.某医⽣有两个预约病⼈，⼀个在下午1点，⽽另⼀个在下午⼀点半。约定的持续时间是均值为30分钟的独⽴指数随机变量。假设两个病⼈都准时到达，求约定在⼀点半的病⼈在医⽣的办公室所花的期望时间。6、(5.38)Let{Mi(t),t>=0},i=1,2,3beindependentPoissonprocesseswithrespectiveratesiλ,andsetN1(t)=M1(t)+M2(t),N2(t)=M2(t)+M3(t),Thestochasticprocess{(N1(t),N2(t)),t>=0}iscalledabivariatePoissonprocess.FindP{N1(t)=n,N2(t)=m}令{Mi(t),t>=0},i=1,2,3是速率分别为iλ（i=1,2,3）的独⽴泊松过程，并且设N1(t)=M1(t)+M2(t),N2(t)=M2(t)+M3(t)。随机过程{(N1(t),N2(t)),t>=0}称为⼆维泊松过程。求P{N1(t)=n,N2(t)=m}。7、(5.42)Let{N(t),t>=0}beaPoissonprocesswithrateλ.LetSndenotethetimeofthenthevent.Find(a)E[S4](b)E[S4|N(1)=2](c)E[N(4)-N(2)|N(1)=3]令{N(t),t>=0}是速率为λ的泊松过程。以Sn记第n个事件发⽣的时间。求(a)E[S4]，(b)E[S4|N(1)=2]，(c)E[N(4)-N(2)|N(1)=3]。8、(5.45)Let{N(t),t>=0}beaPoissonprocesswithrateλ,thatisindependentofthenonnegativerandomvariableTwithmeanµandvariance2σ.Find(a)Cov(T,N(T)),(b)Var(N(T))令{N(t),t>=0}是速率为λ的泊松过程，它独⽴于有均值µ和⽅差2σ的⾮负随机变量T。求(a)Cov(T,N(T)),(b)Var(N(T))。9、(5.47)Consideratwo-serverparallelqueuingsystemwherecustomersarriveaccordingtoaPoissonprocesswithrateλ,andwheretheservicetimesareexponentialwithrateµ.Moreover,supposethatarrivalsfindingbothserversbusyimmediatelydepartwithoutreceivinganyservice(suchacustomerissaidtobelost),whereasthosefindingatleastonefreeserverimmediatelyenterserviceandthendepartwhentheirserviceiscompleted.(a)Ifbothserversarepresentlybusy,findtheexpectedtimeuntilthenextcustomerentersthesystem.(b)Startingempty,findtheexpectedtimeuntilbothserversarebusy.(c)Findtheexpectedtimebetweentwosuccessivelostcustomers.考虑有两条服务线的平⾏排队系统，其中顾客按速率为λ的泊松过程到达，⽽服务时间是速率为µ的指数时间。此外，假设到达者发现两条服务线都忙，就不接受任何服务⽽⽴即离开（这称为顾客流失），只要发现⾄少有⼀条服务线有空，就⽴刻接受服务⽽在服务完成后离开。（a）如果两条服务线现在都忙，求指导第⼆个顾客进⼊系统的平均时间。（b）在开始时系统是空着。求直到两条服务线都忙的平均时间。（c）求相继地两个流失顾客之间的平均时间。10、(5.49)EventsoccuraccordingtoaPoissonprocesswithratλe.Eachtimeaneventoccur,wemustdecidewhetherornottostop,withourobjectivebeingtostopatthelasteventtooccurpriortosomespecifiedtimeT,whereT>1/λ.Thatis,ifaneventoccursattimet,0<=t<=T,andwedecidetostop,thenwewiniftherearenoadditionaleventsbytimeT,andweloseotherwise.IfwedonotstopwhenaneventoccursandnoadditionaleventsoccurbytimeT,thenwelose.Also,ifnoeventsoccurbytimeT,thenwelose.Considerthestrategythatstopsatthefirsteventtooccuraftersomefixedtimes,0<=s<=T.(a)usingthisstrategy,whatistheprobabilityofwinning?(b)Whatvalueofsmaximizestheprobabilityofwinning?(c)Showthatone'sprobabilityofwinningwhenusingtheprecedingstrategywithvalueofsspecifiedinpart(b)is1/e.事件按速率为λ的泊松过程发⽣。在每个事件发⽣的时间，我们必须决定继续还是停⽌，使我们的对象在⼀个特定的时刻T以前在最后的⼀个事件发⽣的时间上停⽌，其中T>1/λ。即如果⼀个事件在时间t（0<=t<=T）发⽣，并且我们决定停⽌，那么若在T之前没有附加事件，则我们赢，否则我们都输。若在⼀个事件发⽣时我们没有停⽌，⽽在T之前⼜没有附加事件，则我们输。此外，若在T之前没有时间，则我们输。考察在⼀个固定的时间s（0<=s<=T）后的⾸个事件发⽣时停⽌的策略。（a）使⽤这个策略时赢的概率是多少？（b）使得赢的概率达到最⼤的s值是多少？（c）证明⼀个⼈在⽤以上的策略，并且按（b）特定的s值时，他赢得概率是1/e。11、(5.53)Thewaterlevelofacertainreservoirisdepletedataconstantrateof1000unitsdaily.Thereservoirifrefilledbyrandomlyoccurringrainfalls.RainfallsoccuraccordingtoaPoissonprocesswithrate0.2perday.Theamountofwateraddedtothereservoirbyarainfallis5000unitswithprobability0.8or8000unitswithprobability0.2.Thepresentwaterlevelisjustslightlybelow5000units.(a)Whatistheprobabilitythereservoirwillbeemptyafterfivedays?(b)Whatistheprobabilitythereservoirwillbeemptysometimewithinthenexttendays.某⽔库的蓄⽔⽔平按每天1000单位的常数速率耗损。⽔库⽔源由随机发⽣的降⾬补给。降⾬按每天0.2的速率的泊松过程发⽣。由⼀次降⾬加进⽔库的⽔量以概率0.8为5000单位，⽽以概率0.2为8000单位。现在的蓄⽔⽔平刚刚稍低于5000单位。（a）在5天后⽔库空的概率是多少？（b）在以后的10天中的某个时间⽔库空的概率是多少？12、(5.55)ConsiderasingleserverqueuingsystemwherecustomersarriveaccordingtoaPoissonprocesswithrateλ,servicetimesareexponentialwithrateµ,andcustomersareservedintheorderoftheirarrival.Supposethatacustomerarrivesandfindsn-1othersinthesystem.LetXdenotethenumberinthesystematthemomentthatthiscustomerdeparts.FindtheprobabilitymassfunctionofX.考虑⼀个单服务线的排队系统，其中顾客按速率为λ的泊松过程到达，服务时间是速率为µ的指数时间，顾客按到达的次序接受服务。假设⼀个顾客到达时发现在系统中有n-1个顾客。以X记这个顾客离开时系统中的⼈数。求X的概率质量函数。13、（每年都考）(5.86)Ingoodyears,stormsoccuraccordingtoaPoissonprocesswithrate3perunittime,whileinotheryearstheyoccuraccordingtoaPoisonprocesswithrate5perunittime.Supposenextyearwillbeagoodyearwithprobability0.3.LetN(t)denotethenumberofstormsduringthefirstttimeunitsofnextyear.(a)FindP{N(t)=n}(b)Is{N(t)}aPoissonprocess?(c)Does{N(t)}havestationaryincrements?Whyorwhynot?(d)Doesithaveindependentincrements?Whyorwhynot?(e)Ifnextyearstartsoffwiththreestormsbytimet=1,whatistheconditionalprobabilityitisagoodyear?在好的年度，暴风⾬按每单位时间速率3的泊松过程发⽣，⽽在其余年度，按每单位时间速率5的泊松过程发⽣。假设明年是好的年度的概率为0.3。以N（t）记明年的前t个单位时间中暴风⾬的次数。（a）求P{N(t)=n}；（b）{N(t)}是泊松过程吗？（c）{N(t)}有没有平稳增量，为什么？（d）它有没有独⽴增量？为什么？（e）如果明年在t=1以前有3次暴风⾬，这是⼀个好的年度的条件概率是多少？14、(5.87)DetermineCov[X(t),X(t+s)]when{X(t),t>=0}isacompoundPoissonprocess.15、(5.88)Customersarriveattheautomaticte