Problem 1 (6-40) The random variables x and y are of discrete type, independent, with P(x = n) = an, P(y = n) = bn, n = 0, 1, . . . Show that, if z = x + y, then P(z = n) = >akbn-k, n =0,1, 2, ... K = 0 Problem 2 (6-43) Let x and y be independent identically distributed nonnegative discrete random variables with P ( x = k) = P(y = k) = Pk k = 0, 1, 2, ... Suppose P ( x = k | x + y = k) = P( x = k - 1/x+y=k)= k + 1 k20 Show that x and y are geometric random variables. (This result is due to Chatterji.) Problem 3 (6-55) Let x represent the number of successes and y the number of failures of n independent Bernoulli trials with p representing the probability of success in any one trial. Find the distribution of z = x - y. Show that E[z] = n(2p - 1), Var(z) = Anp(1 -p). Problem 4 Consider the function (for c > 0) f (x, y ) = (2 - 12, -c