南京大学随机过程练习题附中文解释及答案

4、(2.37)Let X1, X2, …, Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum(X1, X2, …, Xn). Find the distribution function of M. 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。
3、(2.27)A fair coin is independently flipped n times, k times by A and n-k times by B. Find that the probability that A and B flip the same number of heads. 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 {same number of heads} =
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Another argument is as follows. P{# heads of A = # heads of B}= P{# tails of A = # heads of B} since coin is fair = P{k − # heads of A = # heads of B}= P{k = total # heads}.
3、(4.32) Each of two switches is either on or off during a day. On day n, each switch will independently be on with probability [1+#of on switches during day n-1]/4. For instance, if both switches are on during day n-1, then each will independently be on during day n with probability3/4. What fraction of days are both switches on? What fractions are both off? 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？
2、(4.24) Consider three urns, one colored red, one white, and one blue. The red urn contains 1 red and 4 blue balls; the white urn contains 3 white balls, 2 red balls, and 2 blue balls; the blue urn contains 4 white balls, 3 red balls, and 2 blue balls. At the initial stage, a ball is randomly selected from the red urn and then returned to that urn. At every subsequent stage, a ball is randomly selected from the urn whose color is the same as that of the ball previously selected and is then returned to that urn. In the long run, what proportion of the selected balls are red? What proportion are white? What proportion are blue? 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？
5、(2.43)An urn contains n+m balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let X be the number of red balls removed before the first black ball is chosen. We are interested in determining E[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]。）
E[X]=E[X|C1]P(C1)+E[X|C2]P(C2)+...+E[X|Cn]P(Cn)
9、(3.23)A coin having probability p of coming up heads is successively flipped until two of the most recent three flips are heads. Let N denote the number of flips. (Note that if the first two flips are heads, then N = 2). Find E[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]。
pB, where pB>pA. If you objective is to minimize the number of games you need to play to win two in a row, should you start with A or with B? 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？
（以第九版为准） 第二章 Random Variables 随机变量 1、（2.16）An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up? 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？
P {7 games} =
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Differentiation yields that
d P{7} 20[3 p2 (1 p)3 p33(1 p)2 ] = 60 p2 (1 p)2[1 2 p] . dp
Thus, the derivative is zero when p = 1/2. Taking the second derivative shows that the maximum is attained at this value.
答：1（- 0.95）52 - 52*（0.95）51 * 0.05
2、(2.25 略变动)Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1-p. The winner of the series is the first team to win i games. If i = 4, find the probability that a total of 7 games are played. Find the p that maximizes/minimizes this probability. 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：A total of 7 games will be played if the first 6 result in 3 wins and 3 losses. Thus,
1、(4.23) Trials are performed in sequence. If the last two trials were successes, then the next trial is a success with probability 0.8; otherwise the next trial is a success with probability 0.5. In the long run, what proportion of trials are successes? 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？
8、(3.8)An unbiased die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a six and a five. 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]
10、(3.26)You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA; whenever you play B, you win with probability
6、(2.64)Show that the sum of independent identically distributed exponential random variables has a gamma distribution.
证明独立同分布的指数随机变量之和有伽马分布。
7 、 (2.77)Let X and Y be independent normal random variables, each having parameters and 2 . Show that X+Y is independent of X-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。