The transition matrix for a three-state Markov chain with S = {1, 2, 3} is 0 0 1 0 0 1 0.5 0.5 0 P = (a) Show that the chain is irreducible. (b) Show that the chain is periodic and find the period d. Find Pnd, n = 1, 2, .... (c) Use the identity M = E + (M - diag(M))P to find the mean recurrent times. Show that they agree with the formula given in the Basic Limit Theorem for periodic Markov Chains.