Distribution (FDD) of random process Xt

Hello everyone!

The question I have is:

and are indpt standard normal random variables, we have a random process ,where:

I would like to find out the distribution (one dimension FDD) of ,for a fixed

Here is what I did:

by Total Probability Formula

by independence

here is where I stuck...how should I deal with , or is there other ways to this kind of things?

What about n dimensional FDD?

Could anyone help me? :)

Re: Distribution (FDD) of random process Xt

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 :)

Re: Distribution (FDD) of random process Xt

Looks like I am answering my own question again...

After a bit study, I found it is a very simple problem...maybe people won't type so much for a easy Q like this :P

and are indpt standard normal random variables,

therefore , so that

the 1D FDD is

this could be easily extend to nD FDD