For the following statements, either provide a proof that it is true in all cases or construct a counter example:

(a) If X and Y are jointly distributed random variables with Cov(X, Y ) = 0, then X and Y must be independent.

(b) V ar(X Y ) V ar(X Y )

(c) If X and Y are independent then V ar(X Y ) = V ar(X Y ).

(d) Let Y be a binomial random variable with n trials and py probability of success. Let X be another binomial random variable with m trials and px probability of success on each trial. If m > n and E[X] = E[Y ] then V ar(X) > V ar(Y ).