is a random sample from a distribution, where and are unknown.

I'm given:

where

Now, I think that Z is the standard normal dist, and U is dist. Chi Sq with n-1 degrees of freedom.

I'm now asked to find the joint pdf of Z and U; how would I do this?

I'm also given:

and asked to deduce the joint pdf of T and U; again, I think T has the t dist. with n-1 degrees of freedom, but I'm not sure how to go about doing this question.