Suman Thapa

**sumanthapa** · November 13th 2012, 07:21 AM

I need a help to solve the question regarding normal distribution. The question is :
If X and Y are standard normal distribution and are independent then how to find the pdf and mgf of XY?

**abender** · November 13th 2012, 08:17 AM

The (univariate) normal PDF is given by $f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^-{\tfrac{(x-\mu)^2} {2\sigma^2}}$ .

When $X$ is standard normal, $\mu=0 \text{ and } \sigma=1$ , so that the standard normal PDF is $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-{\tfrac{x^2} {2}}}$ .

$Y$ is also a standard normal random variable, so its PDF is similarly $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{-{\tfrac{y^2} {2}}}$ .

Now, If $X$ and $Y$ are INDEPENDENT, then $f_X(x) \cdot f_Y(y) = f_{X,Y}(x,y)$ .

So, $f_{X,Y}(x,y) = f_X(x) f_Y(y) = \left(\frac{1}{\sqrt{2\pi}}e^{-{\tfrac{x^2} {2}}}\right)\left(\frac{1}{\sqrt{2\pi}}e^{-{\tfrac{y^2} {2}}}\right) = \frac{1}{2\pi}e^{-{\tfrac{x^2+y^2} {2}}}$ .

**sumanthapa** · November 13th 2012, 08:42 AM

Thanks for the solution. But I mean, if Z = XY (product of X and Y), where X and Y are standard normal distribution, then how to find the pdf and moment generating function(mgf) of Z? Could you please help me? Thanks.

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