Past exam of the statistics course of the University of Oxford / 2015 / 1 / a / i / Solution

The given probability density function is that of a beta distribution with parameters $r$ and $n-r+1$. We can derive the required moments directly rather than assuming the beta moment formulas. For an integer $j\ge 0$, the Euler beta function givesHere $B(a,b)=(a-1)!(b-1)!/(a+b-1)!$ for positive integers $a,b$; this identity also follows by repeated integration by parts. Taking $j=1,2$ yieldsSubtract the square of the first moment from the second to obtain the variance:These are exact finite-sample identities for each uniform order statistic.