[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be a continuous random variable with density

function [math]f_X(x)[/math]. Show that if

[[math]] \int_{-\infty}^{+\infty} x^2 f_X(x)\, dx \lt \infty\ , [[/math]]

then

[[math]] \int_{-\infty}^{+\infty} |x| f_X(x)\, dx \lt \infty\ . [[/math]]

Hint: Except on the interval [math][-1, 1][/math], the first integrand is greater than the second integrand.