The following related discrete problem also gives a good clue for the answer to Exercise 32. Randomly select with replacement t1, t2, . . . , tr from the set (1/n, 2/n, . . . , n/n). Let X be the smallest value of r satisfying t1 + t2 + €¢ €¢ €¢ + tr > 1 . Then E(X) = (1 + 1/n) n. To prove this, we can just as well choose t1, t2, . . . , tr randomly with replacement from the set (1, 2, . . . , n) and let X be the smallest value of r for which t1 + t2 + €¢ €¢ €¢ + tr > n . (a) Use Exercise 3.2.36 to show that P(X ≥ j + 1) = (n/j) (1/n)j (b) Show that

(c) From these two facts, find an expression for E(X). This proof is due to Harris Schultz.15

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