1. a) Prove that if X is a discrete random variable defined on a finite sample space, S, then we have E(X) = SUM X(a) P({a}).

b) Use part(a) to prove that if X and Y are discrete random variable defined on the same finite sample space S, then we have E(X + Y) = E(X) + E(Y).

2. Use the result of problem 1 above to show that the average number of heads when a fair coin is flipped n times is equal to n/2. Hint: Let H_i be the number of heads on the ith flip. {Note: compare this to the "brute-force" computation to get an interesting combinatorial identity!]