## 本次澳洲代写主要为随机过程相关的限时测试，以下是作业具体内容：

1.（24 分。上传您的手写答案和简短的理由。

(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。

(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？

Short justi cations worth half marks.)

An oce 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 fXngn1 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?