Suppose that X and Y are independent. Prove that for any A R and B R, we have P(X A, Y B) = P(X A)P(Y B). You may make use of the following hints: (i) We learned that if X and Y are independent, then for any functions g and h, E(g(X)h(Y )) = E(g(X)) E(h(Y )). (ii) Let I denote the indicator function. For any A R, we have E(I(X A)) = 1 P(X A) 0 P(X / A) = P(X A)