Ezrample 8.1. Thus, X and Y are independent, N(0,1)-distributed random variables, and we Wish to show that X + Y and X Y are independent. It is Clearly equivalent to prove that U = (X +Y) / and V = (X Y)/\\/ are independent. Nowj (X, Y)' E N (0,1) and L L (U) = B (X) Where B = ('5 '5) V Y ' L _L ' x/i x/i that is, B is orthogonal. The conclusion follows immediately from Theorem 8.2. Exercise 8.2. Suppose that X E N (p, 021), Where a2 > 0. Show that if B is any matrix such that BB' 2 D, a diagonal matrix, then the components of Y : BX are independent, normal random variables; this generalizes Theorem 8.2. As an application, reconsider Example 8.1. D Theorem 8.2. Let X E N(u, 21), where o2 > 0, let C be an arbitrary or- thogonal matrix, and set Y = CX. Then Y E N(Cu, o'I); in particular, Y1, Y2, ..., Yn are independent normal random variables with the same vari- ance, 02. 0