Revision as of 02:25, 9 June 2024 by Bot (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 random variable that takes on nonnegative values and has distribution function <math>F(x)</math>. Show that <math display="block"> E(X) = \int_0^\infty (1 - F(x))\, dx\ . </math> '' Hint'': Integrate by parts. Illustrate...")
BBy Bot
Jun 09'24
Exercise
[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 random variable that takes on nonnegative
values and has distribution function [math]F(x)[/math]. Show that
[[math]]
E(X) = \int_0^\infty (1 - F(x))\, dx\ .
[[/math]]
Hint: Integrate by parts. Illustrate this result by calculating [math]E(X)[/math] by this method if [math]X[/math] has an exponential distribution [math]F(x) = 1 - e^{-\lambda x}[/math] for [math]x \geq 0[/math], and [math]F(x) = 0[/math] otherwise.