1. Let X and Y be independent Geometric random variables, i.e., P(X = k) = P(Y = k) = (1 -p)pk, k = 0, 1, . ... Let U = min {X, Y}, V = max {X, Y } and W = V - U. (a) Compute P(U = i) and P(W = j) for i, j = 0, 1, . ... (b) Show that U and W are independent. (c) A computer is waiting for two flags to be set. Both flags start out not set. For each flag the conditional probability that the flag is set next time given it is not yet set is 0.5. The two flags operate indepen- dently. How many cycles should you expect to wait before both flags are set including the cycle on which the last flag becomes set