The product of exponential and a uniform random variables

[MathBubble]

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