南京大学随机过程练习题附中文解释及答案.pdf

南南京京⼤⼤学学随随机机过过程程练练习习题题附附中中⽂⽂解解释释及及答答案案

（以第九版为准）

第⼆章RandomVariables随机变量

1（2.16）Anairlineknowsthat5percentofthepeoplemakingreservationsonacertainflightwillnotshowup.Conseuently,theirpolicyisto

sell52ticketsforaflightthatcanholdonly50passengers.Whatistheprobabilitythattherewillbeaseatavailableforeverypassengerwho

showsup?

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,eachofwhichisindependentlywonbyteamAwithprobabilitypandby

teamBwithprobability1-p.Thewinneroftheseriesisthefirstteamtowinigames.Ifi=4,findtheprobabilitythatatotalof7gamesare

played.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{2332ppppPdp

d==]21[)1(6022ppp--.Thus,th

ederivativeiszerowhenp=1/2.Takingthesecondderivativeshowsthatthemaximumisattainedatthisvalue.

3(2.27)Afaircoinisindependentlyflippedntimes,ktimesbyAandn-ktimesby

B.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).Findthe

distribut