answer all questions with explanation

Problem 5 Entropy a) (10 pts) Suppose that we have a horse race with eight horses taking part. Assume that the probabilities of winning for the eight horses are (1/2, 1/4, 1/8, 1/16, 1/64, 1/64, 1/64, 1/64). What is the entropy of the horse race in bits? b) (5 pts*, Grad only) Prove that if X and Y are independent RVs, then H( X, Y) = H(X) + H(Y). It suffices to prove this for discrete RVs. c) (10 pts*, Grad only) Let X,Y be the results of two independent coin flips (H=1, T=0). Let Z = X +Y. Find H(X, Y), H(X|Y), and I(Z; X)