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Math Help - 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 X, Y IID N(0,1), obtain the pdf for 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 R=\sqrt{X^2+Y^2} and looking for the joint pdf of U and R, but the function is not 1-1. Since Z=\frac{Y}{X} is a standard cauchy, I've also tried breaking U into: -\frac{1}{\sqrt{1+Z^2}} for x<0, 0 for x=0, \frac{1}{\sqrt{1+Z^2}} for 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 U^2 (by finding the joint pdf of U^2 and 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 X, Y IID N(0,1), obtain the pdf for 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 R=\sqrt{X^2+Y^2} and looking for the joint pdf of U and R, but the function is not 1-1. Since Z=\frac{Y}{X} is a standard cauchy, I've also tried breaking U into: -\frac{1}{\sqrt{1+Z^2}} for x<0, 0 for x=0, \frac{1}{\sqrt{1+Z^2}} for 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.
    $$ U is obviously the projection onto the x-axis of a uniform RV on the unit circle.

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

    Can you take it from there?

    CB
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