In this question, you will use the MGF to show that the binomial distribution

converges to the Poisson distribution as n goes to infinity.

(a) (3 pts) Y1, ..., Yn are iid Poisson distributed random variable with the variance of np, and define Y = Y1 +...+Yn. Please find the MGF of Y, namely MY (t) for t R.

(b) (3 pts) X1,...,Xn are iid Bernoulli(np ) random variables, and define X = X1 + ... + Xn. Computes the MGF of X1 and X, namely MX1 (t) and MX (t) for t R.

(c) (4 pts) Prove that as n , the MGF of X converges to the MGF of Y , namely MX (t) MY (t) for t R.