(a) Let X and Y be any two random variables. Prove that E(max{X, Y }) = E(X) + E(Y ) E(min{X, Y }).

Interestingly, this is analogous to the probability law P(A B) = P(A) + P(B) P(A B).

(b) A random variable X is said to have a discrete uniform distribution on the set {a,a+1,...,b} if

P(X =x)= 1 , x=a,a+1,...,b. ba+1

Find the moment generating function of X.