MULTIVARIATE DISTRIBUTION

QUESTION 1 (a) If an n x m matrix X is N (1u'; In 8 ) , derive the density function of X. (b) Show that if A is Wm (n, _) with n > m, then the density function of A is fA (A) = 2-7mn ( det )) zn I'm (n) etr ([A) (det A) =( -m-1) , A > 0 2m n 2mn where Vm,n ( HidH1) I'm (n) (c) If A is Wm (n, ) with n > m - 1, show that E [(det A)" ] = (det >)" 2mr I'm (in +7) I'm (n) (d) If A is Wm (n, _) show that the characteristic function of tr A is E [etr (it A)] = det (Im - 2it )) in and for m = 2 E [etr (itA)] = [(1 - 2it)1) (1 - 2itA2)]in where 1 and 12 are the latent roots of _ (e) If A is Wm (n, _) where n 2 m, prove that det A/ det ) is distributed as the product of m independent x2 random variables with degrees of freedom n, n - 1, ..., n -m +1, respectively