88. In Section 3.6.3, we saw that if U is a random variable that is uniform on (0, 1)

and if, conditional on U = p, X is binomial with parameters n and p, then P{X = i} =

1 n + 1

, i = 0, 1,..., n For another way of showing this result, let U, X1, X2,..., Xn be independent uniform (0, 1) random variables. Define X by X = #i: Xi < U That is, if the n + 1 variables are ordered from smallest to largest, then U would be in position X + 1.

(a) What is P{X = i}?

(b) Explain how this proves the result of Section 3.6.3.