1) X, Y and Z are random vectors. Show that: (a) Var(X)=E(XX)E(X)E(X). (b) Cov(X,Y)=(Cov(Y,X)). (c) Var(X+Y)=Var(X)+Var(Y)+Cov(X,Y)+Cov(Y,X).

( X' stands for X- transpose and so on)

2) We say that X is χ2(Chi-square) distributed with n degrees of freedom (denoted as X ∼ χ2n) if X can be written as X = Z12 + ... + Zn2, where Z1,...,Zn are independent standard normal random variables. Show that EX = n and Var(X) = 2n. Hint: Use the moment generating function to show that if Z ∼ N(0,1), then EZ4 = 3