1 Problem .1 Let X and Y be two random variables (discrete or continuous, pick). Prove the following statements: a E(doX + 1) = doE(X) + 1 where do,1 are real numbers. b E ( X + Y ) = E(X ) + E( Y ). c Var(doX + )1) = AgVar(X) where do,1 are real numbers. d Var (X) = E(X2) - E(X)2 e Cov( X, Y) = E(XY ) - E(X) E( Y) f Var ( X +Y) = Var(X) + Var(Y) +2Cou(X, Y) g If X and Y are independent E(X |Y = y) = E(X) h E ( YX| Y = y) = yE(X| Y = y) i Law of iterated expectations. E(E(X |Y) ) = E(X)