(i) Suppose U1,..., Un are i.i.d. U(0, 1) and let U(k) denote the kth largest value (or kth order statistic). Find the density of U(k) and show that P{U(k) ≤ p} = p 0

n!

(k − 1)!(n − k)!

uk−1

(1 − u)

n−k du , which in turn is equal to n

j=k n

j p j

(1 − p)

n− j .

(ii) Use (i) to show that, in Example 3.5.2 with 0 < x < n, the Clopper-Pearson solution pˆU for an upper 1 − α confidence bound for p can be expressed as the 1 − α quantile of the Beta distribution with parameters x + 1 and n − x.