1. [6 points total, 2 points each] Let X be a continuous random variable with expected value E(X) and the probability density function = f(x) = e c a for x>0 otherwise. for constants a > 0 and c> 0. (a) Find a and c. (b) Compute the variance Var(X). (c) Let Y = x=3. Compute the standard deviation (X + Y). 2. [6 points total, 2 points each] Suppose that X and Y are random variables such that E(X)=3, E(Y) = 1, Var(X) = 2 and Cov(X + Y, X - Y) = 1. Find the following: (a) Var(X/3) (b) E(X+2Y) (c) Var(Y) 3. [8 points total, 4 points each] (a) Let c be a constant and X a random variable with expected value E(X) and variance Var(X). Show that E[(X - c)] = Var(X) + (c E(X)). = (b) Suppose that X and Y are random variables with expected values E(X) E(Y) = 0, variances Var(X) and Var(Y) and covariance Cov(X, Y). Show that Var(XY) Var(X)Var(Y) = Cov(X, Y) [Cov(X, Y)].