Exercise 3.17. Let Y Uniform([0, 1]) and X Binomial(n,Y ). (i) Use iterated expectation for probability to write P(X = k) = n k !Z 1 0 y k (1 y) nk d y. (1) (ii) Write An,k = R 1 0 y k (1 y) nk d y. Use integration by parts and show that An,k = k n k +1 An,k1. (2) for all 1 k n. Conclude that for all 0 k n, An,k = 1 n k 1 n +1 . (3) (iii) Conclude that X Uniform({0, 1, ,n})

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