Grinstead-Snell-Probability · Problem 6.3.14

Sums of Random Variables

Mathematics · expected-value-variance-continuous

Problem

Let XX be a continuous random variable with density function fX(x)f_X(x). Show that if ∫−∞+∞x2fX(x) dx<∞ ,\int_{-\infty}^{+\infty} x^2 f_X(x)\,dx < \infty\,, then ∫−∞+∞∣x∣fX(x) dx<∞ .\int_{-\infty}^{+\infty} |x| f_X(x)\,dx < \infty\,. Hint: Except on the interval [−1,1][-1, 1], the first integrand is greater than the second integrand.

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