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

1. Jun 18, 2012

### MathBubble

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. Jun 18, 2012

### Ray Vickson

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

RGV