Do the following: For random variable X, show that var(X) = E[X2] - E[X]2. (Hint: start with var(X) = E[(X - E[X])2]). For random variables X and Y, show that cov(X, Y) = E[XY] - E[X]E[Y]. (Hint: start with cov (X, Y) = E[(X - E[X]) (Y - E[Y])]). For random variables X and Y, show that var (X + Y) = var (X) + var(Y)+2 cov (X, Y). (Hint: start with var(X+Y) = E[(X - E[X] + Y - E[Y])2]). Solution a). Var(X) = E([X ? E(X)]^2 ) = E(X ^2 ? 2XE[X] + E[X]^2 ) = E[X ^2] ? 2E[X]E[X] + E[X]^2 = E[X ^2 ] ? E[X]^2 b). Cov(X, Y ) = E([X ? E(X)][Y ? E(Y )]) = E(XY ? XE[Y ] ? E[X]Y + E[X]E[Y ]) = E[XY ] ? E[X]E[Y] ? E[X]E[Y ] + E[X]E[Y] = E(XY ) ? E(X)E(Y ) c). Var (X + Y ) = E [(X + Y )^2] ? (E [X + Y ])^2 = E[ X^2 + 2XY + Y^2] ? (E [X] + E [Y ])^2 = E[ X^2] + 2E [XY ] + E[ Y^2] ?{ (E [X])^2+ 2E [X] E [Y ] + (E [Y ])^2} = E [X^2] ? (E [X])^2 + E[ Y^2] ? (E [Y ])^2 + 2E [XY ] ? 2E [X] E [Y ] = Var (X) + Var (Y ) + 2E [XY ] ? 2E [X] E [Y ] = Var (X) + Var (Y ) +2 Cov(X, Y ).