Problems

The random variables \(X_1\), \(X_2\), \(\ldots\) , \(X_{2n+1}\) are independently and uniformly distributed on the interval \(0 \le x \le 1\). The random variable \(Y\) is defined to be the median of \(X_1\), \(X_2\), \(\ldots\) , \(X_{2n+1}\). Given that the probability density function of \(Y\) is \(\g(y)\), where \[ \mathrm{g}(y)=\begin{cases} ky^{n}(1-y)^{n} & \mbox{ if }0\leqslant y\leqslant1\\ 0 & \mbox{ otherwise} \end{cases} \] use the result $$ \int_0^1 {y^{r}}{{(1-y)}^{s}}\,\d y = \frac{r!s!}{(r+s+1)!} $$ to show that \(k={(2n+1)!}/{{(n!)}^2}\), and evaluate \(\E(Y)\) and \({\rm Var}\,(Y)\). Hence show that, for any given positive number \(d\), the inequality $$ {\P\left({\vert {Y - 1/2} \vert} < {d/{\sqrt {n}}} \right)} < {\P\left({\vert {{\bar X} - 1/2} \vert} < {d/{\sqrt {n}}} \right)} $$ holds provided \(n\) is large enough, where \({\bar X}\) is the mean of \(X_1\), \(X_2\), \(\ldots\) , \(X_{2n+1}\). [You may assume that \(Y\) and \(\bar X\) are normally distributed for large \(n\).]