Consider a reservoir with a total capacity of h units of water. Let X,, represent the amount of water in the reservoir at the end of the n-th day. You are given the following information.. The daily inputs to the reservoir are independent and discrete random variables. On a given day, P(input is j units ) = gift, j = 0, 1,2,.... . Any overflow (when the total amount of water exceeds the capacity of h units) is regarded as a loss. . Provided the reservoir is not empty, one unit is released at the end of the day. . The value of X, for day n is the content of the reservoir after the release at the end of the day. Therefore, the stochastic process { X,, n = 1, 2, ...} is a Markov chain with state space {0, 1, 2, ... . h- 1}. (a) Explain why the stochastic process is a Markov Chain. (b) Show that it is irreducible, aperiodic and positive recurrent. For the remaining part of this question, assume h = 3. (c) Determine the one-step transition matrix. (d) Let 7; be the first hitting time of state i. Find P(To