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这是一篇作业主要为随机过程相关的限时测试
1.（24 分。上传您的手写答案和简短的理由。
简短的理由值半分。）
一名办公室工作人员拥有 3 把雨伞，她用来从家里去上班，反之亦然
反之。如果早上下雨，她会带一把雨伞（如果她有的话）
去上班，如果晚上下雨，她会带一把雨伞（如果她有的话）
她的回家之旅。如果不下雨，她就不带伞。一开始
在每次旅行中，下雨的概率为 p = 1 q，与之前的旅行无关。
令 Xn 表示第 n 个开始时可用的雨伞数量
并假设 0 < p < 1。 (a) 解释为什么 fXngn 1 是状态空间 S = f0 的马尔可夫链； 1; 2; 3克 具有转移概率矩阵 P： (b) 对 (a) 中马尔可夫链的状态进行分类。 (c) 直接解释或计算为什么 P(X3 = 0 j X1 = 0) = 1 p。 (d) 令 = ( 0; 1; 2; 3) 是 (a) 中马尔可夫链的平稳分布。 假设我们已经证明了 1 = 2 = 3。找到 0 的值； 1; 2 和 3。 (e) 如果第一次旅行开始时没有雨伞可用，计算 平均旅行次数，直到再次没有雨伞可用。 (f) 解释为什么 P(Xn = i) 的极限为 n ！ 1 存在，因为 i = 0； 1; 2; 3;和 计算它们。 (g) 经过大量旅行后，平均有多少雨伞可用 旅行的开始 (h) 在大量的旅行中，她得到的旅行比例大约是多少 湿的 1. (24 marks. Upload your handwritten answers and short justications. Short justications worth half marks.) An oce worker owns 3 umbrellas which she uses to go from home to work and vice versa. If it is raining in the morning she takes an umbrella (if she has one) on her trip to work and if it is raining at night she takes an umbrella (if she has one) on her trip home. If it is not raining, she doesn’t take an umbrella. At the beginning of each trip it is raining with probability p = 1 q, independently of previous trips. Let Xn represent the number of umbrellas available at the beginning of the n-th trip and assume that 0 < p < 1. (a) Explain why fXngn1 is a Markov Chain with state space S = f0; 1; 2; 3g having the transition probability matrix P: (b) Classify the states of the Markov chain in (a). (c) Explain or calculate directly why P(X3 = 0 j X1 = 0) = 1 p. (d) Let = (0; 1; 2; 3) be a stationary distribution of the Markov chain in (a). Suppose we have proved 1 = 2 = 3. Find the values of 0; 1; 2 and 3. (e) If no umbrellas are available at the beginning of the rst trip, calculate the average number of trips until there are again no umbrellas available. (f) Explain why the limits of P(Xn = i) as n ! 1 exists, for i = 0; 1; 2; 3; and calculate them. (g) After a large number of trips, how many umbrellas are available on average at the beginning of a trip (h) In a large number of trips, about what proportion of journeys does she get wet