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:

$\displaystyle p_x(\alpha) =\left\{ \begin{array}{rl}

\frac1{\pi \sqrt{1 - \alpha^2}} &\mbox{ if $-1 \leq \alpha \leq 1$} \\

0 &\mbox{ otherwise}

\end{array} \right.$

$\displaystyle p_y(\beta) =\left\{ \begin{array}{rl}

\beta e^{- \beta^2} &\mbox{ if $\beta \geq 0$} \\

0 &\mbox{ otherwise}

\end{array} \right.$

Show that $\displaystyle z = xy$ has a Gaussian density function.

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I tried solving it by manipulating it like this:

Let $\displaystyle y = \beta$ where $\displaystyle \beta$ is fixed. Then the random variable x is transformed to $\displaystyle z=\beta x$. We then, know that

$\displaystyle p_z(\alpha) = \frac1{|\beta|} p_x \left(\frac{\alpha}{\beta}\right)$

Now $\displaystyle p_z(\gamma) = \int_{-\infty}^{\infty} \frac1{|\beta|} p_{x,y} \left(\frac{\gamma}{\beta}, \beta \right) \, d\beta$

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)