54. 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 = ñ, X is binomial with parameters ç and p, then For another way of showing this result, let U, ×ß9×2, ...,Xn be independent uniform (0, 1) random variables. Define X by That is, if the ç + 1 variables are ordered from smallest to largest, then U would be in position X + 1.

(a) What is P[X = /}?

(b) Explain how this proves the result stated in the preceding.