Question: 17 Let X be a continuous random variable. Prove that P(|X|n) E(XI) 1+ P(|X|n). These important inequalities show that E(X) < if and only if
17 Let X be a continuous random variable. Prove that P(|X|n) E(XI) 1+ P(|X|n). These important inequalities show that E(X) < if and only if the series P(Xn) converges. Hint: By Exercise 16, *** E(X) = P(X)>t) = P(\X\ > t) dt = * * * P (|X|>t)dt. Note that on the interval [n, n+1), P(Xn+1)P(|X|>t) P(|X|n).
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