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Math Help - 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 {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
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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 {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}<x<\sqrt{y_0+\delta/2}) + P(-\sqrt{y_0+\delta/2}<x<-\sqrt{y_0-\delta/2})

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