Let $X_1,\ldots,X_n\sim U[0,1]$ be i.i.d. uniform random variables and let $X_{(k)} $ denote the $k$'th smallest variable.

Given some $\tau\in(0,1)$, what is $$\mathbb E[X_{(k)}\mid X_{(k)}\le \tau]?$$

Can we give a simple-to-use lower bound if the exact expression is not simple?

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    $\begingroup$ It is well-known that the order statistics of uniform has a beta distribution, therefore you are asking the mean of a truncated beta, which may involve incomplete beta I think. $\endgroup$ – BGM Nov 14 at 17:12
  • $\begingroup$ @BGM - thanks for the comment. Do you know if there are any simple lower bounds on the expectation of the truncated beta distribution? $\endgroup$ – R B Nov 15 at 13:59

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