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On the fourth Probability example sheet of this term the following rather cute result appeared as a question:

Let $X,Y,Z$ be uniform random variables on $(0,1)$. Then $(XY)^Z$ is also uniform on $(0,1)$.

This seems so unlikely to be true, but somehow, if you work through the integrals, it is. If someone can give me a really good reason I’d love to hear it. That taking logs of a uniform distribution gives a scaled exponential distribution of parameter 1 was vaguely useful, but didn’t quite completely explain the whole problem.

I shall not post my proof of this fact. It is not pretty, is fairly routine, and is on a tripos example sheet so I shouldn’t give it away.