exercise:7a776abb92: 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 with cumulative distribution function <math>F</math> strictly increasing on the range of <math>X</math>. Let <math>Y = F(X)</math>. Show that <math>Y</math> is uniformly distributed in the interval <math>[...") |
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Let <math>X</math> be a random variable with cumulative distribution function <math>F</math> strictly increasing on the range of <math>X</math>. Let <math>Y = F(X)</math>. Show that <math>Y</math> is uniformly distributed in the interval <math>[0,1]</math>. (The formula <math>X = F^{-1}(Y)</math> then tells us how to construct <math>X</math> from a uniform random variable <math>Y</math>.) | |||
strictly increasing on the range of <math>X</math>. Let <math>Y = F(X)</math>. Show that <math>Y</math> is uniformly | |||
distributed in the interval <math>[0,1]</math>. (The formula <math>X = F^{-1}(Y)</math> then tells us how | |||
to construct <math>X</math> from a uniform random variable <math>Y</math>.) | |||
Latest revision as of 02:05, 14 June 2024
Let [math]X[/math] be a random variable with cumulative distribution function [math]F[/math] strictly increasing on the range of [math]X[/math]. Let [math]Y = F(X)[/math]. Show that [math]Y[/math] is uniformly distributed in the interval [math][0,1][/math]. (The formula [math]X = F^{-1}(Y)[/math] then tells us how to construct [math]X[/math] from a uniform random variable [math]Y[/math].)