Let (Sn)n>o be a simple random walk defined by p(1) = p and p(-1) = q where p + q = 1. Let k be a positive integer. (a) Prove that Po(S1 > 0, . .., S2k-1 > 0, S2k = 0) = pq Po( S1 2 0, . .., S2k-3 2 0, S2k-2 = 0). (b) By using the formula for Catalan numbers (Corollary 17 of the lecture notes), prove that 2k - 1 Po(S1 > 0, . . . , S2k-1 > 0, S2k = 0) = 2k-1 k-1 phqh. (c) Hence use part (b) to prove the following result: Po(T. = 2k | S2k = 0) = 2k - 1' where To = inf{n 2 1 : Sn =0}