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