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Math Help - Joint Distribution, expected value correlation of a graphed triangle (Fixed)

  1. #1
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    Lightbulb Joint Distribution, expected value correlation of a graphed triangle (Fixed)

    Suppose X and Y are two independent random variables each distributed as $Uniform(0,1)$.

    1) Find the joint distribution of X and Y.

    2) Let U = \cos(2 \pi Y)\sqrt{-2\ln(X)} and $V = \sin(2\pi Y)\sqrt{-2\ln(X)}$. Find the joint distribution
    of U and V assuming that the transformation is one-to-one?

    3) Find the marginal distributions of U and V?

    For 1, I got that my distribution is 1 by multiplying the two distributions together. For 2, I have been getting stuck trying to get my equations in terms of X and Y to perform the transformation. I simplified and got Y = (\frac{1}{2})$ \tan^{-1}(\frac{V}{U})$ but something seems wrong about this and am having trouble trying to get X. For 3, I am obviously stuck and even not totally sure on the support. Any help is appreciated.
    Last edited by WUrunner; December 14th 2012 at 09:32 PM.
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  2. #2
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    Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)

    Hey WUrunner (and thanks for fixing the latex up).

    For V you can use the transformation theorem in probability and for U you can write things in terms of V.

    Also is sin(2) meant to be sin(2*pi*Y)? If not then the tangent expression will be wrong.
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  3. #3
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    Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)

    Thanks for pointing out my mistake. I somehow missed my sin function
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  4. #4
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    Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)

    Also you want to look at ratio distributions to get the distribution of a ratio of two independent variables:

    Ratio distribution - Wikipedia, the free encyclopedia
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