1. Let X ~ Pois()) and Y ~ Pois() be independent. (a) Write a formula for the moment generating function Mx(t) of X as an infinite sum. (b) Evaluating the infinite sum, we get Mx (t) = ex(et-1) and similarly My (t) = em(et-1). Using this information, identify the distribution of X + Y. 2. (a) Find the moment generating function of a Bern(p) random variable. (b) Find the moment generating function of a Bin(n, p) random variable. (Hint: If X ~ Bin(n, p), we can write X = Xi+ . .. + Xn where the X; are iid Bern(p).) (c) If M(t) is the answer to part (b), compute M'(t) and use this to verify that the expected value of a Bin(n, p) random variable is np (which you knew already)