exercise:Ef1ed87dcb: Difference between revisions
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(Created page with "<div class="d-none"><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></div> Let <math>X</math> be a continuous random variable with density function <math>f_X(x)</math>. Show that if <math display="block"> \int_{-\infty}^{+\infty} x^2 f_X(x)\, dx < \infty\ , </math> then <math display="block"> \int_{-\infty}^{+\infty...") |
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Let <math>X</math> be a continuous random variable with density function <math>f_X(x)</math>. Show that if | |||
function <math>f_X(x)</math>. Show that if | |||
<math display="block"> | <math display="block"> | ||
Latest revision as of 22:38, 14 June 2024
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.