exercise:A118f9b610: 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 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...") |
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Let <math>X</math> be a random variable that takes on nonnegative values and has distribution function <math>F(x)</math>. Show that | |||
values and has distribution function <math>F(x)</math>. Show that | |||
<math display="block"> | <math display="block"> | ||
Latest revision as of 22:38, 14 June 2024
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.