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Thread: Change of variables when the function is not 1-1

  1. #1
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    Change of variables when the function is not 1-1

    I ran into this problem:
    Suppose $\displaystyle X, Y IID N(0,1)$, obtain the pdf for $\displaystyle U=\frac{X}{\sqrt{X^2+Y^2}}$.
    Because of the nasty square root, I can't seem to find the appropriate transformation functions for a change of variables. I tried letting $\displaystyle R=\sqrt{X^2+Y^2}$ and looking for the joint pdf of $\displaystyle U$ and $\displaystyle R$, but the function is not 1-1. Since $\displaystyle Z=\frac{Y}{X}$ is a standard cauchy, I've also tried breaking $\displaystyle U$ into: $\displaystyle -\frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x<0, 0$ for $\displaystyle x=0, \frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x>0$, but that doesn't help much.
    What am I supposed to do when I see this type of questions? I need some hints.
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  2. #2
    Moo
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    Hello,

    I'm giving a quick thought. Can't you first find the pdf of $\displaystyle U^2$ (by finding the joint pdf of $\displaystyle U^2$ and $\displaystyle R^2$) ? Don't forget changing the boundaries, as you have symmetry for x and for y.
    I don't know what method you use, so please show a little work and I may be more able to help you !
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Crescent View Post
    I ran into this problem:
    Suppose $\displaystyle X, Y IID N(0,1)$, obtain the pdf for $\displaystyle U=\frac{X}{\sqrt{X^2+Y^2}}$.
    Because of the nasty square root, I can't seem to find the appropriate transformation functions for a change of variables. I tried letting $\displaystyle R=\sqrt{X^2+Y^2}$ and looking for the joint pdf of $\displaystyle U$ and $\displaystyle R$, but the function is not 1-1. Since $\displaystyle Z=\frac{Y}{X}$ is a standard cauchy, I've also tried breaking $\displaystyle U$ into: $\displaystyle -\frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x<0, 0$ for $\displaystyle x=0, \frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x>0$, but that doesn't help much.
    What am I supposed to do when I see this type of questions? I need some hints.
    $\displaystyle $$ U$ is obviously the projection onto the x-axis of a uniform RV on the unit circle.

    That is $\displaystyle $$ U$ has the same distribution as $\displaystyle X=\cos(A),\ A\sim U(0,2\pi)$

    Can you take it from there?

    CB
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