[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[(Xc)2]=Var(X)+(cE(X))2. (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(X2,Y2)[Cov(X,Y)]2..

1. [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[(Xc)2]=Var(X)+(cE(X))2. (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(X2,Y2)[Cov(X,Y)]2.