Let (, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance.

(a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent.

(b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated

(i.e. Cov(W, Z) > 0).

(a) Your solution

(b) Your solution