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?

  • 2
    $\begingroup$ Is there a probability $\frac1n$ that $U_1=x$? $\endgroup$
    – Henry
    Jun 25, 2021 at 15:58
  • $\begingroup$ 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. $\endgroup$ Jun 25, 2021 at 16:32
  • $\begingroup$ I'm asking for a single $U_i$ but the other situation is also interesting $\endgroup$
    – wayne
    Jun 25, 2021 at 16:38
  • 1
    $\begingroup$ Discussion of a special case in these answers: math.stackexchange.com/a/1007778/321264, math.stackexchange.com/a/843061/321264. $\endgroup$ Jun 25, 2021 at 18:11
  • $\begingroup$ @StubbornAtom Thank you! $\endgroup$
    – wayne
    Jun 26, 2021 at 10:32


You must log in to answer this question.