南京大学随机过程练习题附中文解释及答案

（以第九版为准）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置的概率是多少？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},{=i)2/1()2/1(knkiknik=i)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，…，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,eachhavingparametersand2.ShowthatX+YisindependentofX-Y.假设X和Y是独立正态随机变量，都具有均值和方差2。证明X+Y与X-Y独立。8、(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]。重要：E[E[X|Y]]=E[X]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,wherepBpA.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pBpA。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，成功的比例是多少？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,andthenumberoftimeperiodsth