A random variable X has the probability distribution function f (x) = 1 3

for x = −1, 0, or 1.

(a) Find M[X; t], the moment generating function for X.

(b) If X1 and X2 are independent observations of X, find M[X1 + X2; t] without first finding the probability distribution of X1 + X2.

(c) Verify the result in part

(b) by finding the probability distribution of X1 + X2.