The discussion focuses on proving that the random variable Z = U(X+Y) has the same distribution as X, where X and Y are independent random variables following an exponential distribution with parameter λ, and U is uniformly distributed on (0,1). It is established that X+Y follows a gamma distribution, specifically gamma(2,λ). The solution involves computing the Laplace transform of Z, denoted as \(\tilde{Z}(s) \equiv Ee^{-sZ}\), and conditioning on U to facilitate the analysis.
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MathBubble said: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.