Two statistical random variables combine to give Gaussian?

Hi all,

Sorry for such a strange title. The question I am stuck at is...

Let x and y be statistically independent random variables with p.d.fs:

Show that has a Gaussian density function.

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I tried solving it by manipulating it like this:

Let where is fixed. Then the random variable x is transformed to . We then, know that

Now

Following this I used Independence Property and wrote the joint pdf as the product of pdfs. However the resulting integral was too strange. I could not solve it(I am intimidated by it). So I think there must be some other ways to do it.

So if anyone has any idea on the problem, please enlighten me.(Doh)

Thanks,

Iso

**P.S**: I wonder whether I should have posted the integral in the Calculus forum (Thinking)