1. Assume Xn are independent and identically distributed with P(X1 = 1) = p, P(X1 = 0) = r and P(X1 = -1) = q. where p, r, q > 0 and p + r + q = 1. Let Sn = En Xi, n = 1, 2, .... (a) Prove that {S1, S2, ...} is an irreducible Markov chain with state space S = {0, +1, 12, ...} and write down its transition matrix. (b) Is the chain aperiodic? (c) Find expressions for: i. P(S3 = 2). ii. P(SA = 1\\S1 = 1). iii. P(S10 = 1|$7 = 0). iv. ES, and var(Sn)