I found a similar Q with answer, just post it on if it helps...

and are indpt standard normal random variables

Find the n dimensional FDD.

and the answer looks like:

if ,

if ,

I understand where the comes from, take 2D case, like above:

again using TPF:

but how does this becomes

???

I believe the previous should have the similar FDD, so could anyone explain this to me??

Thank you for your help