Let Y1, . . . , Yn be iid Uniform(0, 1) random variables. Let Y(i) be the ith order statistic.

1. (a) Derive the probability density function of Y(i).
2. (b) For 1 i < j n, derive the joint density of (Y(i), Y(j)).
3. p
4. (c) Show Y(n) 1 in probability.
5. (d) Show that n(1Y(n)) converge in distribution to a limit. Identify the limiting distribution.