Are min(X,Y) and X-Y independent given X>Y in an exponential distribution?

[Chris L T521]

Here's this week's problem.

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Problem: Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Argue that, conditional on $X>Y$, the random variables $\min(X,Y)$ and $X-Y$ are independent.

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[Chris L T521]

No one answered this week's problem. I haven't completed the solution yet (almost done), but I'll update this post later today with a solution. Sorry about that!