Question: 18. Let X be a continuous random variable. Prove that . n=1 P ! |X| n E ! |X| 1

18. Let X be a continuous random variable. Prove that .∞ n=1 P ! |X| ≥ n " ≤ E ! |X| " ≤ 1 +.∞ n=1 P ! |X| ≥ n " . These important inequalities show that E ! |X| " / < ∞ if and only if the series ∞ n=1 P ! |X| ≥ n " converges. Hint: By Exercise 17, E ! |X| " = E ∞ 0 P ! |X| > t" dt = .∞ n=0 E n+1 n P ! |X| > t" dt. Note that on the interval [n, n + 1), P ! |X| ≥ n + 1 " < P! |X| > t" ≤ P ! |X| ≥ n " .

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