# Math Help - random process (Gaussian distribution)

1. ## random process (Gaussian distribution)

Hi,

Well as you see I’m a newbie..
This was one of the problems on my telecommunications exam but it’s pure math:

Assuming that X is a random Gaussian variable and ${Y}=X^2$,find $f_{Y}\left({y}\right)$

$f_{X}\left({x}\right)=\frac{1}{\sqrt{2\pi}\sigma}e ^{\frac{{-}x^2}{2\sigma^2}}$

any help appreciated

2. Originally Posted by Rezast
Hi,

Well as you see I’m a newbie..
This was one of the problems on my telecommunications exam but it’s pure math:

Assuming that X is a random Gaussian variable and ${Y}=X^2$,find $f_{Y}\left({y}\right)$

$f_{X}\left({x}\right)=\frac{1}{\sqrt{2\pi}\sigma}e ^{\frac{{-}x^2}{2\sigma^2}}$

any help appreciated
One rather clunky way of doing it is this:

if $y \ne 0$

$P(y_0-\delta/2 < y < y_0+\delta/2)=P(y_0-\delta/2 < x^2 < y_0+\delta/2)$

..... $= P(\sqrt{y_0-\delta/2}

..... $=2 P(\sqrt{y_0-\delta/2}

(this last since the pdf of X is symmetric)

..... $=2 P(\sqrt{y_0}-(1/4)\delta/ \sqrt{y_0}

..... $\approx 2 p_X(\sqrt{y_0}) \left( \frac{1}{2}\, \frac{\delta}{\sqrt{y_0}}\right)$

..... $= p_X(\sqrt{y_0}) \left( \frac{\delta}{\sqrt{y_0}}\right)$

..... $= p_Y(y_0) \delta$

So we have:

$p_Y(y_0)=\frac{p_X(\sqrt{y_0})}{\sqrt{y_0}}$

So the density of $Y$ is:

$p_Y(y)=\frac{p_X(\sqrt{y})}{\sqrt{y}}$,

where $p_X(x)$ is the density of $X$.

(I'm pretty sure that this has gone wrong somewhere, but you should get the idea)

RonL