exercise:A944d87d2d: Difference between revisions

Latest revision as of 01:23, 14 June 2024

Let [math]X_1[/math] and [math]X_2[/math] be independent random variables and for [math]i = 1, 2[/math], let [math]Y_i = \phi_i(X_i)[/math], where [math]\phi_i[/math] is strictly increasing on the range of [math]X_i[/math]. Show that [math]Y_1[/math] and [math]Y_2[/math] are independent. Note that the same result is true without the assumption that the [math]\phi_i[/math]'s are strictly increasing, but the proof is more difficult.

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-<div class="d-none"><math>
+Let <math>X_1</math> and <math>X_2</math> be independent random variables and for
-\newcommand{\NA}{{\rm NA}}
+<math>i = 1, 2</math>,  let
-\newcommand{\mat}[1]{{\bf#1}}
+<math>Y_i = \phi_i(X_i)</math>, where <math>\phi_i</math> is strictly increasing on the range of
-\newcommand{\exref}[1]{\ref{##1}}
+<math>X_i</math>.  Show that <math>Y_1</math> and <math>Y_2</math> are independent.  Note that the same result is true
-\newcommand{\secstoprocess}{\all}
+without the assumption that the <math>\phi_i</math>'s are strictly increasing, but the proof is
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mathds}{\mathbb}</math></div> Let </math>X_1<math> and </math>X_2<math> be independent random variables and for
-</math>i = 1, 2<math>,  let
-</math>Y_i = \phi_i(X_i)<math>, where </math>\phi_i<math> is strictly increasing on the range of
-</math>X_i<math>.  Show that </math>Y_1<math> and </math>Y_2<math> are independent.  Note that the same result is true
-without the assumption that the </math>\phi_i$'s are strictly increasing, but the proof is
 more difficult.