Let $U_1,U_2,\dots,U_n$ be i.i.d. uniform random variables on the interval $[a,b]$. Set $Z := \max\{U_1,U_2,\dots,U_n\}$. How can I determine the conditional distribution $\mathbb{P}(U_i \mid Z = x)$. I think it must be the uniform distribution on the interval $[a,x]$. If so, how can I prove this formally?
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StubbornAtom
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wayne
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2Is there a probability $\frac1n$ that $U_1=x$? – Henry Jun 25 '21 at 15:58
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Are you asking the question for a single $U_i$ or for all at the same time? Answer by NN2 seems to be for all at once. – herb steinberg Jun 25 '21 at 16:32
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I'm asking for a single $U_i$ but the other situation is also interesting – wayne Jun 25 '21 at 16:38
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1Discussion of a special case in these answers: https://math.stackexchange.com/a/1007778/321264, https://math.stackexchange.com/a/843061/321264. – StubbornAtom Jun 25 '21 at 18:11
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@StubbornAtom Thank you! – wayne Jun 26 '21 at 10:32