The product of exponential and a uniform random variables

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Homework Statement

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.

Homework Equations

The Attempt at a Solution

X+Y is gamma(2,λ) distributed. But I can't figure out how to deal with this product.

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

X+Y is gamma(2,λ) distributed. But I can't figure out how to deal with this product.

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

RGV