STOCHASTIC

(e) Show that X has mean and variance equal to
{exp0*+l/20
are constants (c = e°). Show that c TliX?' ~ A(.a + Y.b,Mi, S>iW)-
(j) Show that if X~ Afeff 2 ), its reciprocal 1/X~ A{-n,a 2 ).
(k) Suppose X, ~ A(nt, a 2 ) are independent, i = 1,2. Show that
X1IX2~ AUn-H^oS + Oi 2 ).
It is often of interest to consider moment distributions of log-normal variates.
(1) Suppose X ~ AO.cr 2 ), and Xs is they'th moment of X about the origin. Show that
{$%X> dA{X;n,c 2 ))jXj ~ A{x:n+jc 2 ,a 2 ).
21. (Properties of multivariate log-normal distributions) Suppose Yi,...,Ym have the
multivariate normal distribution N(/x,E), where ft is the mean vector and £ is the mxn
symmetric positive-definite variance-covariance matrix. Then the vector X' = (.Xi,..., X,„) =
(exp Yit..., exp Ym) has the multivariate log-normal distribution A (X; u, Z).
(a) Show that the density of X is
where logX= (logJT1,...,logA'm)'.
As in the univariate case, the study of moments of X is facilitated by using the momentgenerating
function of Y. For any real vector f = (ri,..., t,„), the mgf of a random m-vector Z
is ,M0 = E[e' z l
(b) Let j = (Ji,... ,jm)'. Show that the y'th cross moment of Z about the origin equals
8Ei,^(0)
8fi',...,ee'
(c) Show that the variance of w,, the ith component of Z, equals
e 2 ^z(0) 8Vz(0)
354 PART III STATIC PORTFOLIO SELECTION MODELS
0',