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

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.

For odd sample size, the sample median is the central order statistic, $M_U=U_{(m+1)}$. Since $n=2m+1$, its index is $r=(n+1)/2$ and $n-r+1=(n+1)/2$. Substitute these into the expectation and variance formulas from part (i):In fact $M_U$ has the symmetric beta distribution $\mathrm{Beta}(m+1,m+1)$, explaining why its expectation is exactly the centre of the interval.