MULTIVARIATE DISTRIBUTION THEORY

QUESTION 1 Suppose X is Em (0, V) where I is diagonal. If X1, ... I'm are all independent prove that X is normally distributed. QUESTION 2 (a) Suppose that A is Wm (n, E) and X : N x m is N (0, IN Q E) (n 2 m) where A and X are independent. For B = A + XX find the joint density function of B and X. (b) Let A and B be independent, where A is Wm (n1, Im) and B is Wm (n2, Im) with m > m - 1 and n2 > m- 1 Derive the density function of U = A ?BA-1. (c) Derive the joint density function of the latent roots u1, ..., Um of U in 4(b) 1 2m2 m where f (u1, ..., Um) = I'm (am) g (U1, ..., Um) II (uj - uj) i