1. Let &, k > 0, be independent random variables taking only the integers. Let S = {0, +1, 12, ..., IN}, where N = co is allowed. Let Xo be another random variable, independent of the sequence , taking values in S and let f : S x Z -> S be a certain real function. Define new random variables X, by Xntl = f(Xn, {n), n = 0, 1, 2... (a) Show that {X,} , is a Markov chain with state space S and this chain is homogeneous if only &, k 2 1, have the same distribution. (b) Suppose that Xo = 0, f(x, y) = r +y and {, k > 0, are iid with P(60 = 1) = p, P(60 = 0) = r and P(60 = -1) = q. where p, r, q > 0 and ptr +q =1. i. Write down the transition probabilities of the chain {X}21. ii. Is the chain {X,}>, aperiodic and irreducible? iii. Find expressions for: P(X: = 2), P(X, = 1/X, =1) and P(X10 = 1/X7 = 0)