1. The random vector (X, Y )t has a two-dimensional normal distribution with Var X = Var Y . Show that X + Y and X - Y are independent random variables.
2. Suppose that X and Y have a joint normal distribution with E X = E Y = 0, Var X = σ2 , Var Y = σ2, and correlation coefficient ρ. Compute E XY x y and Var XY.
Remark. One may use the fact that X and a suitable linear combination of X and Y are independent.