Find (a) the expectation and (b) the variance of the kth-order statistic of a sample of...

Find (a) the expectation and (b) the variance of the kth-order statistic of a sample of n independent random variables uniform on [0, 1]. The density function is given in Example C in Section 3.7.

Reference

For the case where the X_i are uniform on [0, 1], the density of the kth-order statistic reduces to

This is the beta density. An interesting by-product of this result is that since the density integrates to 1,