- #1

Torgny

I have proved (8.1). However I am trying to prove that

##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function:

##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})##

I am trying to use the moment generating function because there is only one moment generating function for a given probability distribution and this also holds for multivariate distributions. But I fail at obtaining a moment generating function. The mgf to ##E(e^{tX_i})## is simply the mgf to the normal distribution but I cant get a moment generating function to ##E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## which from the answer in the text i guess should be a multivariate normal distribution.

Can someone help out?