1
$\begingroup$

enter image description here

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?

$\endgroup$
1
$\begingroup$

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: $$A\mathbf X=(\overline X,X_1-\overline X,\dots,X_n-\overline X)^T$$

$\endgroup$
  • $\begingroup$ It seems like a very nice answer! Unfortunately I have trouble grasping the fundamentals here. Thanks! $\endgroup$ – torgny Oct 26 '17 at 20:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.