# Expected value of the $k$'th order statistic given that it's smaller than $\tau$

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?

• 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. – BGM Nov 14 at 17:12
• @BGM - thanks for the comment. Do you know if there are any simple lower bounds on the expectation of the truncated beta distribution? – R B Nov 15 at 13:59