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

I've been able to show that $X+Y$ has a $\Gamma(2, \lambda)$ distribution, but how do I calculate the distribution of this product?

To clarify: The answer by Sasha works, but I was looking for a solution that does something like integrate over a suitable area.

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