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Homework Help: The product of exponential and a uniform random variables

  1. Jun 18, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to show that U(X+Y) = X in distribution, where X and Y are independent exp(λ) distributed and U is uniformly distributed on (0,1) independent of X+Y.

    2. Relevant equations

    3. The attempt at a solution
    X+Y is gamma(2,λ) distributed. But I can't figure out how to deal with this product.
  2. jcsd
  3. Jun 18, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Get the distribution of [itex]Z = U(X+Y)[/itex] by computing its Laplace transform
    [tex]\tilde{Z}(s) \equiv Ee^{-sZ}.[/tex]
    Hint: condition on U.

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