N is a discrete random variable taking values on the positive integers and

P(N = n) = (3/4)(1/4)^(n1) for n {1, 2, . . .}.

Also, for each i, Xi is an independent Bernoulli random variable with P(Xi = 1) = 1/4 and P(Xi = 0) = 3/4. Define the random variable S = X_1 + + X_N .

Find P(S = 0)

Find the generating function of X, which is the Bernoulli random variable. Hence, find the generating function of S, i.e. find E[t^S] .