<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Characteristic function</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Characteristic_function"> <link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Characteristic_function rootpage-Characteristic_function skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Characteristic function</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the term "<b>characteristic function</b>" can refer to any of several distinct concepts:
</p>
<ul><li>The <a href="Indicator_function" title="Indicator function">indicator function</a> of a <a href="Subset" title="Subset">subset</a>, that is the <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo>:<!-- : --></mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}</annotation>
</semantics>
</math></span></span> which for a given subset <i>A</i> of <i>X</i>, has value 1 at points of <i>A</i> and 0 at points of <i>X</i> − <i>A</i>.</li>
<li>The <a href="Characteristic_function_(convex_analysis)" title="Characteristic function (convex analysis)">characteristic function</a> in <a href="Convex_analysis" title="Convex analysis">convex analysis</a>, closely related to the indicator function of a set: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>0</mn>
<mo>,</mo>
</mtd>
<mtd>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>A</mi>
<mo>;</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>+</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo>,</mo>
</mtd>
<mtd>
<mi>x</mi>
<mo>∉</mo>
<mi>A</mi>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}</annotation>
</semantics>
</math></span></span></li>
<li>In <a href="Probability_theory" title="Probability theory">probability theory</a>, the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of any <a href="Probability_distribution" title="Probability distribution">probability distribution</a> on the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> is given by the following formula, where <i>X</i> is any <a href="Random_variable" title="Random variable">random variable</a> with the distribution in question: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>t</mi>
<mi>X</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}</annotation>
</semantics>
</math></span></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} }</annotation>
</semantics>
</math></span><img src="./92e9e770e56a457072dc96780ee6e2f207eb21f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} }" loading="lazy"></span> denotes <a href="Expected_value" title="Expected value">expected value</a>. For <a href="Joint_probability_distribution" title="Joint probability distribution">multivariate distributions</a>, the product <i>tX</i> is replaced by a <a href="Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of vectors.</li>
<li>The characteristic function of a <a href="Cooperative_game_theory" title="Cooperative game theory">cooperative game</a> in <a href="Game_theory" title="Game theory">game theory</a>.</li>
<li>The <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> in <a href="Linear_algebra" title="Linear algebra">linear algebra</a>.</li>
<li>The <a href="Characteristic_state_function" title="Characteristic state function">characteristic state function</a> in <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>.</li>
<li>The <a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a>, a <a href="Topology" title="Topology">topological</a> invariant.</li>
<li>The <a href="Receiver_operating_characteristic" title="Receiver operating characteristic">receiver operating characteristic</a> in statistical <a href="Decision_theory" title="Decision theory">decision theory</a>.</li>
<li>The point characteristic function in <a href="Statistics" title="Statistics">statistics</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */
.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}
/* end https://en.wikipedia.org/ */
</style><div class="reflist">
</div>
<p><br>
<style data-mw-deduplicate="TemplateStyles:r1097304697">
/* start https://en.wikipedia.org/ */
.mw-parser-output .dmbox{display:flex;align-items:center;clear:both;margin:0.9em 1em;border-top:1px solid #ccc;border-bottom:1px solid #ccc;padding:0.25em 0.35em;font-style:italic}.mw-parser-output .dmbox>*{flex-shrink:0;margin:0 0.25em}.mw-parser-output .dmbox-body{flex-grow:1;flex-shrink:1;padding:0.1em 0}.mw-parser-output .dmbox-invalid-type{text-align:center}
/* end https://en.wikipedia.org/ */
</style>
</p>
</div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2024-03-06" href="https://en.wikipedia.org/wiki/?title=Characteristic_function&oldid=1212221433">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>