Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 42 / 1 / ii / Solution

This is a symmetric beta distribution with parameters $(2,2)$, so its mean and median are both $1/2$. At the median its probability density function is $f(1/2)=3/2$, giving asymptotic variance $1/9$ for the sample median. Direct integration givesConsequently $\sqrt{n}({\bar{X}}_n-1/2)\stackrel{d}{\to }N(0,1/20)$. The sample median has $20/9$ times the asymptotic variance of the sample mean, with asymptotic relative efficiency