Suppose that X1, . . . , Xm and Y1, . . . , Yn are two independent samples with Xi N(1, 2 1 ) and Yi N(2, 2). The difference between the sample means X and Y is then a linear combination of m + n normal random variables.

(a) (3 points). Find E(X Y ).

(b) (3 points). Find var(X Y ).

(c) (5 points). Use moment generating functions to show that the distribution of X Y is normal.