proving a multivariate normal distribution by the moment generating function

Question

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?

Answer 1

Hint:

If $\mathbf X=(X_1,\dots, X_n)^T$ has a normal distribution then it is well known that also $A\mathbf X$ has a normal distribution when $A$ is a matrix having $n$ columns.

So it is enough to find a matrix $A$ that satisfies: $$A\mathbf X=(\overline X,X_1-\overline X,\dots,X_n-\overline X)^T$$