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Thread: random process (Gaussian distribution)

  1. #1
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    random process (Gaussian distribution)

    Hi,

    Well as you see Iím a newbie..
    This was one of the problems on my telecommunications exam but itís pure math:

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

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

    any help appreciated
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  2. #2
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    Quote Originally Posted by Rezast View Post
    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 $\displaystyle {Y}=X^2$,find $\displaystyle f_{Y}\left({y}\right)$

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

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

    if $\displaystyle y \ne 0$

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

    .....$\displaystyle = P(\sqrt{y_0-\delta/2}<x<\sqrt{y_0+\delta/2}) + P(-\sqrt{y_0+\delta/2}<x<-\sqrt{y_0-\delta/2}) $

    .....$\displaystyle =2 P(\sqrt{y_0-\delta/2}<x<\sqrt{y_0+\delta/2}) $

    (this last since the pdf of X is symmetric)


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

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

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

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

    So we have:

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

    So the density of $\displaystyle Y$ is:

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

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

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

    RonL
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