8.1 Let Y Nn(????, ) follow a multivariate normal distribution and set Xi = exp(Yi ),...

8.1 Let Y ∼ Nn(????, Σ) follow a multivariate normal distribution and set Xi =

exp(Yi

), i = 1, ..., n. The purpose of this exercise is to find the moments of X. They are most easily calculated using the moment generating function for Y M(u) = E

{

exp (

uTY

)} = exp (

uT???? +

1 2

uTΣu

)

as a function of u = (u1, ..., un)

T.

Let ei denote an n-vector with a one in the ith place and zeros elsewhere.

Show that E(Xi

) = M(ei

) and E(Xi Xj

) = M(ei + ej

), i, j = 1, ..., n, and hence verify the moments in (8.1)–(8.2).