I've got a question about the distribution of the random variable $Y = X_{1}X_{2}$, where $X_i \sim \operatorname{N}(0,1)$, and $X_1$, $X_2$ independent.  Specifically, I need to find the MGF of $Y$.  Here's what I've tried:  The $r^\text{th}$ moment of $Y$ is given by:  $$\begin{align} \operatorname{E} (Y^r) &amp; = \operatorname{E} (X_1^r X_2^r) \\ &amp; = \operatorname{E} (X_1^r) \operatorname{E} (X_2^r) &amp; \text{because } X_1 \text{ and } X_2 \text{ are independent} \\ \end{align}$$  Since $X_i \sim \operatorname{N}(0,1)$, I know that the $r^\text{th}$ central moment is given by:  $$ \operatorname{E}(X_i^r) = \begin{cases} 0       &amp; r \text{ odd} \\ (2r-1)! &amp; r \text{ even} \\ \end{cases}$$  Because $\operatorname{E}(X_i^r)$ does not depend on $i$, $\operatorname{E} (X_1^r) = \operatorname{E} (X_2^r)$ and so  $$\operatorname{E} (X_1^r) \operatorname{E} (X_2^r) = ((2r-1)!)^2 \quad \text{for } r \text{ even}$$  But I can't really derive a formula for the $r^{\text{th}}$ integral, especially if I don't know how the formula for $\operatorname{E}(X_i^r)$ looks before plugging in $t=0$.   Are there other ways to derive the MGF of $Y$?   EDIT:  I also need to show that $Y$ can be expressed as a difference of two independent, Gamma-distributed variables.
We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ?
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