To preface, I know some basic calculus but not a lot about random variables. But I'm curious about the situation where $X$ is $ U(0,1)$ and $Y$ is $U(0,X)$. Is it possible to get the distribution or expected value of Y, and if so, is there a way to handle random variables with random parameters in general?
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2Yes, condition on $X$ – Andrew Apr 14 '23 at 03:55
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If $Z\sim U(0,1)$ is independent of $X$ then $Y=XZ.$ – Gérard Letac Apr 14 '23 at 06:44