Please do the following exercises, showing all of your work. 1. Let X and Y be two independent random variables with densities fx (x) and fy(x). a) What is the joint density fix,ry(x, y)? b) Optional Extra Credit: Establish the following: for two bounded functions g and h, E(g( X)h(Y)) = E(g(X)) E(h(Y)) 2. Let X be a random variable and b be a constant. Show that X and b are independent 3. Show that in general Var(X + Y) = Var(X") + 2Cov(X, Y) + Vary 4. Show that if X&Y are independent, then Cov(X, Y) = 0 5. Use the previous problems to show that if X and Y are independent, then Var(X + Y) = Var(X") + Var(Y) 6. Show that Var(a X + b) = a' Var(X), where b is a constant. Let X and Y have a joint density given by f(x, y) = 2 0