(i) If the r.v.s X and Y are independent, distributed as N(0, (2), and U = X + Y, V = X - Y, then show that U and V are independent, distributed as N(0, 2(2), by transforming the joint p.d.f of X and Y into the joint p.d.f. of X and Y into the joint p.d.f. of U and V.
(ii) If the r.v.s X and Y are independent, distributed N((1, (2) and N ((2, (2), respectively, use part (i) in order to show that U and V are independent, distributed as N((1 + (2, 2(2) and N ((1 - (2, 2(2), respectively?