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Math Help - Distributions, transformations and marginals

  1. #1
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    Distributions, transformations and marginals

    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 $U = \cos(2\piY)\sqrt{-2\ln(X)}$ and $V = \sin(2\piY)\sqrt{-2\ln(X)}$. 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.
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  2. #2
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    Re: Distributions, transformations and marginals

    Hey WUrunner.

    Can you please fix up your latex or post the equation in non-latex form? It's a little hard to decipher as it is shown.
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