7. (Problem 3.2.1 in Pinsky and Karlin) Consider the Markov chain whose transition probability matrix is given by 0.3 0.4 0.2 0.1 0.1 0.3 0.4 0.2 P = 0.4 0.2 0.1 0.3 0.2 0.1 0.3 0.4 Suppose that the mitial distribution is pi = 0.25 for i = 0, 1, 2,3. Use proof by induction to show that Pr(Xn = k) = 0.25, k = 0, 1, 2,3, for all n. Comment