Micron Document
<!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>Multivariate random variable</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Multivariate_random_variable"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<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-Multivariate_random_variable rootpage-Multivariate_random_variable 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">Multivariate random variable</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"><style data-mw-deduplicate="TemplateStyles:r1129693374">
/* start https://en.wikipedia.org/ */


.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}


/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1246091330">
/* start https://en.wikipedia.org/ */


.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}


/* end https://en.wikipedia.org/ */
</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">
<ul><li><a href="Probability" title="Probability">Probability</a>
<ul><li><a href="Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>
<li><a href="Determinism" title="Determinism">Determinism</a>
<ul><li><a href="Deterministic_system" title="Deterministic system">System</a></li></ul></li>
<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>
<li><a href="Randomness" title="Randomness">Randomness</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li>
<li><a href="Sample_space" title="Sample space">Sample space</a></li>
<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event</a>
<ul><li><a href="Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>
<li><a href="Elementary_event" title="Elementary event">Elementary event</a></li>
<li><a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>
<li><a href="Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>
<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>
<li><a href="Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>
<ul><li><a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>
<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a>
<ul><li><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>
<li><a href="Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>
<li><a href="Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>
<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>
<li><a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>
<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>
<li><a href="Probability_measure" title="Probability measure">Probability measure</a></li>
<li><a href="Random_variable" title="Random variable">Random variable</a>
<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>
<li><a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>
<li><a href="Expected_value" title="Expected value">Expected value</a></li>
<li><a href="Variance" title="Variance">Variance</a></li>
<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>
<li><a href="Realization_(probability)" title="Realization (probability)">Observed value</a></li>
<li><a href="Random_walk" title="Random walk">Random walk</a></li>
<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Complementary_event" title="Complementary event">Complementary event</a></li>
<li><a href="Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>
<li><a href="Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>
<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>
<li><a href="Conditional_independence" title="Conditional independence">Conditional independence</a></li>
<li><a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>
<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>
<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>
<li><a href="Boole's_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li>
<li><a href="Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>
</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
/* start https://en.wikipedia.org/ */


.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}


/* end https://en.wikipedia.org/ */
</style></td></tr></tbody></table>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
/* start https://en.wikipedia.org/ */


.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}


/* end https://en.wikipedia.org/ */
</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Multivariate_statistics" title="Multivariate statistics">Multivariate statistics</a>.</div>
<p>In <a href="Probability_theory" title="Probability theory">probability</a>, and <a href="Statistics" title="Statistics">statistics</a>, a <b>multivariate random variable</b> or <b>random vector</b> is a list or <a href="Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vector</a> of mathematical <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual <a href="Statistical_unit" title="Statistical unit">statistical unit</a>. For example, while a given person has a specific age, height and weight, the representation of these features of <i>an unspecified person</i> from within a group would be a random vector. Normally each element of a random vector is a <a href="Real_number" title="Real number">real number</a>.
</p><p>Random vectors are often used as the underlying implementation of various types of aggregate <a href="Random_variable" title="Random variable">random variables</a>, e.g. a <a href="Random_matrix" title="Random matrix">random matrix</a>, <a href="Random_tree" title="Random tree">random tree</a>, <a href="Random_sequence" title="Random sequence">random sequence</a>, <a href="Stochastic_process" title="Stochastic process">stochastic process</a>, etc.
</p><p>Formally, a multivariate random variable is a <a href="Column_vector" class="mw-redirect" title="Column vector">column vector</a> <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 \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">T</mi>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}</annotation>
</semantics>
</math></span><img src="./ada5a9bc21cd37cf25c8cfb2fd2408efa3cd09a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.578ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}" loading="lazy"></span> (or its <a href="Transpose" title="Transpose">transpose</a>, which is a <a href="Row_vector" class="mw-redirect" title="Row vector">row vector</a>) whose components are <a href="Random_variable" title="Random variable">random variables</a> on the <a href="Probability_space" title="Probability space">probability space</a> <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 (\Omega ,{\mathcal {F}},P)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
<mo>,</mo>
<mi>P</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>
</semantics>
</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></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 \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is the <a href="Sample_space" title="Sample space">sample space</a>, <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 {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>
</semantics>
</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is the <a href="Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> (the collection of all events), and <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 P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is the <a href="Probability_measure" title="Probability measure">probability measure</a> (a function returning each event's <a href="Probability" title="Probability">probability</a>).
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Probability_distribution">Probability distribution</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multivariate_probability_distribution" class="mw-redirect" title="Multivariate probability distribution">Multivariate probability distribution</a></div>
<p>Every random vector gives rise to a probability measure on <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 \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> with the <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a> as the underlying sigma-algebra. This measure is also known as the <a href="Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a>, the joint distribution, or the multivariate distribution of the random vector.
</p><p>The <a href="Probability_distribution" title="Probability distribution">distributions</a> of each of the component random variables <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 X_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>
</semantics>
</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> are called <a href="Marginal_distribution" title="Marginal distribution">marginal distributions</a>. The <a href="Conditional_probability_distribution" title="Conditional probability distribution">conditional probability distribution</a> of <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 X_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>
</semantics>
</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> given <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 X_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation>
</semantics>
</math></span><img src="./ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}" loading="lazy"></span> is the probability distribution of <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 X_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>
</semantics>
</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> when <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 X_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation>
</semantics>
</math></span><img src="./ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}" loading="lazy"></span> is known to be a particular value.
</p><p>The <b>cumulative distribution function</b> <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 F_{\mathbf {X} }:\mathbb {R} ^{n}\mapsto [0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }:\mathbb {R} ^{n}\mapsto [0,1]}</annotation>
</semantics>
</math></span><img src="./5b30f84c300818779899583633faa44ef4cc9100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.255ex; height:2.843ex;" alt="{\displaystyle F_{\mathbf {X} }:\mathbb {R} ^{n}\mapsto [0,1]}" loading="lazy"></span> of a random vector <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 \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">T</mi>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}</annotation>
</semantics>
</math></span><img src="./ada5a9bc21cd37cf25c8cfb2fd2408efa3cd09a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.578ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}" loading="lazy"></span> is defined as<sup id="cite_ref-Gallager_1-0" class="reference"><a href="#cite_note-Gallager-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.15">: p.15 </span></sup>
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<style data-mw-deduplicate="TemplateStyles:r1266403038">
/* start https://en.wikipedia.org/ */


.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}


/* end https://en.wikipedia.org/ */
</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})}</annotation>
</semantics>
</math></span><img src="./13ad8b4311ebf247c4854daeb4b4c76dbcbfc00e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.294ex; height:2.843ex;" alt="{\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.1" class="reference nourlexpansion" style="font-weight:bold;">Eq.1</span></td></tr></tbody></table>
</div>
<p>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 \mathbf {x} =(x_{1},\dots ,x_{n})^{\mathsf {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">T</mi>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})^{\mathsf {T}}}</annotation>
</semantics>
</math></span><img src="./65bfe6b31f3dd0da617d72d142c1ece00a5b9093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.781ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})^{\mathsf {T}}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Operations_on_random_vectors">Operations on random vectors</h2></div>
<p>Random vectors can be subjected to the same kinds of <a href="Euclidean_vector#Basic_properties" title="Euclidean vector">algebraic operations</a> as can non-random vectors: addition, subtraction, multiplication by a <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, and the taking of <a href="Dot_product" title="Dot product">inner products</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Affine_transformations">Affine transformations</h3></div>
<p>Similarly, a new random vector <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> can be defined by applying an <a href="Affine_transformation" title="Affine transformation">affine transformation</a> <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 g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:<!-- : --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./6f147e7d7a52fb587d2eb46a4a767351ecfe6c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.557ex; height:2.676ex;" alt="{\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}" loading="lazy"></span> to a random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span>:
</p>
<dl><dd><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 \mathbf {Y} =\mathbf {A} \mathbf {X} +b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>+</mo>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =\mathbf {A} \mathbf {X} +b}</annotation>
</semantics>
</math></span><img src="./351eda808c74d82a5e4865c90c3210defa645b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.995ex; height:2.343ex;" alt="{\displaystyle \mathbf {Y} =\mathbf {A} \mathbf {X} +b}" loading="lazy"></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 \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> is an <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> matrix and <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 b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> is an <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 n\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation>
</semantics>
</math></span><img src="./d24148f103e1cccb60addeeb0a64cb1c3d5622e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}" loading="lazy"></span> column vector.</dd></dl>
<p>If <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 \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> is an <a href="Invertible_matrix" title="Invertible matrix">invertible matrix</a> and <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 \textstyle \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./1b8c6feeb921a6d81808b98802b5caee9ba103a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \textstyle \mathbf {X} }" loading="lazy"></span> has a probability density function <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 f_{\mathbf {X} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\mathbf {X} }}</annotation>
</semantics>
</math></span><img src="./51cce7149cdbfef66c1b70ccfcc069a5031e330c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.799ex; height:2.509ex;" alt="{\displaystyle f_{\mathbf {X} }}" loading="lazy"></span>, then the probability density of <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> is
</p>
<dl><dd><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 f_{\mathbf {Y} }(y)={\frac {f_{\mathbf {X} }(\mathbf {A} ^{-1}(y-b))}{|\det \mathbf {A} |}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo movablelimits="true" form="prefix">det</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\mathbf {Y} }(y)={\frac {f_{\mathbf {X} }(\mathbf {A} ^{-1}(y-b))}{|\det \mathbf {A} |}}}</annotation>
</semantics>
</math></span><img src="./3306fba2ce1fd82b8ef6e95ee0946a678b221cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.463ex; height:6.843ex;" alt="{\displaystyle f_{\mathbf {Y} }(y)={\frac {f_{\mathbf {X} }(\mathbf {A} ^{-1}(y-b))}{|\det \mathbf {A} |}}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Invertible_mappings">Invertible mappings</h3></div>
<p>More generally we can study invertible mappings of random vectors.<sup id="cite_ref-Lapidoth_2-0" class="reference"><a href="#cite_note-Lapidoth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.284–285">: p.284–285 </span></sup>
</p><p>Let <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 g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> be a one-to-one mapping from an open subset <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 {\mathcal {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>
</semantics>
</math></span><img src="./3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"></span> of <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 \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> onto a subset <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 {\mathcal {R}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">R</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation>
</semantics>
</math></span><img src="./74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"></span> of <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 \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>, let <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 g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> have continuous partial derivatives in <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 {\mathcal {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>
</semantics>
</math></span><img src="./3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"></span> and let the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian determinant</a> <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 \det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">det</mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)}</annotation>
</semantics>
</math></span><img src="./ca2be9fa1f57044829e2532b271129a19f0f52f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.603ex; height:6.176ex;" alt="{\displaystyle \det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)}" loading="lazy"></span> of <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 g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> be zero at no point of <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 {\mathcal {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>
</semantics>
</math></span><img src="./3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"></span>. Assume that the real random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> has a probability density function <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 f_{\mathbf {X} }(\mathbf {x} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\mathbf {X} }(\mathbf {x} )}</annotation>
</semantics>
</math></span><img src="./7bdd0c4f093a3b1ad94514b03d24663825a311bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.02ex; height:2.843ex;" alt="{\displaystyle f_{\mathbf {X} }(\mathbf {x} )}" loading="lazy"></span> and satisfies <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 P(\mathbf {X} \in {\mathcal {D}})=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(\mathbf {X} \in {\mathcal {D}})=1}</annotation>
</semantics>
</math></span><img src="./c2a72fea4609d49280a2a257f7fc0ece78f6162e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.468ex; height:2.843ex;" alt="{\displaystyle P(\mathbf {X} \in {\mathcal {D}})=1}" loading="lazy"></span>. Then the random vector <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 \mathbf {Y} =g(\mathbf {X} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =g(\mathbf {X} )}</annotation>
</semantics>
</math></span><img src="./a87935f4907978af41f9d3e2a3e7f9c64911b97c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.063ex; height:2.843ex;" alt="{\displaystyle \mathbf {Y} =g(\mathbf {X} )}" loading="lazy"></span> is of probability density
</p>
<dl><dd><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 \left.f_{\mathbf {Y} }(\mathbf {y} )={\frac {f_{\mathbf {X} }(\mathbf {x} )}{\left|\det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)\right|}}\right|_{\mathbf {x} =g^{-1}(\mathbf {y} )}\mathbf {1} (\mathbf {y} \in R_{\mathbf {Y} })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mo movablelimits="true" form="prefix">det</mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>∈<!-- ∈ --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.f_{\mathbf {Y} }(\mathbf {y} )={\frac {f_{\mathbf {X} }(\mathbf {x} )}{\left|\det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)\right|}}\right|_{\mathbf {x} =g^{-1}(\mathbf {y} )}\mathbf {1} (\mathbf {y} \in R_{\mathbf {Y} })}</annotation>
</semantics>
</math></span><img src="./6fc74eb8270fa1cecbfc3dbc7c28f66760dc42d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:39.321ex; height:10.676ex;" alt="{\displaystyle \left.f_{\mathbf {Y} }(\mathbf {y} )={\frac {f_{\mathbf {X} }(\mathbf {x} )}{\left|\det \left({\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}\right)\right|}}\right|_{\mathbf {x} =g^{-1}(\mathbf {y} )}\mathbf {1} (\mathbf {y} \in R_{\mathbf {Y} })}" loading="lazy"></span></dd></dl>
<p>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 \mathbf {1} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation>
</semantics>
</math></span><img src="./235ffc0f1788b720aef5caa7b97246a84421fd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }" loading="lazy"></span> denotes the <a href="Indicator_function" title="Indicator function">indicator function</a> and set <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 R_{\mathbf {Y} }=\{\mathbf {y} =g(\mathbf {x} ):f_{\mathbf {X} }(\mathbf {x} )>0\}\subseteq {\mathcal {R}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>:</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>&gt;</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>⊆<!-- ⊆ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">R</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{\mathbf {Y} }=\{\mathbf {y} =g(\mathbf {x} ):f_{\mathbf {X} }(\mathbf {x} )&gt;0\}\subseteq {\mathcal {R}}}</annotation>
</semantics>
</math></span><img src="./c2d2501f21cde86999c8614608b1eabd14ea13dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.98ex; height:2.843ex;" alt="{\displaystyle R_{\mathbf {Y} }=\{\mathbf {y} =g(\mathbf {x} ):f_{\mathbf {X} }(\mathbf {x} )>0\}\subseteq {\mathcal {R}}}" loading="lazy"></span> denotes support of <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Expected_value">Expected value</h2></div>
<p>The <a href="Expected_value" title="Expected value">expected value</a> or mean of a random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> is a fixed vector <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} [\mathbf {X} ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {X} ]}</annotation>
</semantics>
</math></span><img src="./3027b4ba8627bb297a8039b230b2afb29a717422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.896ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\mathbf {X} ]}" loading="lazy"></span> whose elements are the expected values of the respective random variables.<sup id="cite_ref-Gubner_3-0" class="reference"><a href="#cite_note-Gubner-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.333">: p.333 </span></sup>
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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} [\mathbf {X} ]=(\operatorname {E} [X_{1}],...,\operatorname {E} [X_{n}])^{\mathrm {T} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">T</mi>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {X} ]=(\operatorname {E} [X_{1}],...,\operatorname {E} [X_{n}])^{\mathrm {T} }}</annotation>
</semantics>
</math></span><img src="./fb2b77d329025627999ff8a4ac19c3dcb12c8af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.267ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [\mathbf {X} ]=(\operatorname {E} [X_{1}],...,\operatorname {E} [X_{n}])^{\mathrm {T} }}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.2" class="reference nourlexpansion" style="font-weight:bold;">Eq.2</span></td></tr></tbody></table>
</div>
<div class="mw-heading mw-heading2"><h2 id="Covariance_and_cross-covariance">Covariance and cross-covariance</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3></div>
<p>The <b><a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a></b> (also called <b>second central moment</b> or variance-covariance matrix) of an <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 n\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation>
</semantics>
</math></span><img src="./d24148f103e1cccb60addeeb0a64cb1c3d5622e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}" loading="lazy"></span> random vector is an <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> whose (<i>i,j</i>)<sup>th</sup> element is the <a href="Covariance" title="Covariance">covariance</a> between the <i>i</i><sup> th</sup> and the <i>j</i><sup> th</sup> random variables. The covariance matrix is the expected value, element by element, of the <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> matrix <a href="Matrix_multiplication" title="Matrix multiplication">computed as</a> <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 [\mathbf {X} -\operatorname {E} [\mathbf {X} ]][\mathbf {X} -\operatorname {E} [\mathbf {X} ]]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\mathbf {X} -\operatorname {E} [\mathbf {X} ]][\mathbf {X} -\operatorname {E} [\mathbf {X} ]]^{T}}</annotation>
</semantics>
</math></span><img src="./26928071db5f676c48f2cf3cdbd4bf86d06f052f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.488ex; height:3.176ex;" alt="{\displaystyle [\mathbf {X} -\operatorname {E} [\mathbf {X} ]][\mathbf {X} -\operatorname {E} [\mathbf {X} ]]^{T}}" loading="lazy"></span>, where the superscript T refers to the transpose of the indicated vector:<sup id="cite_ref-Lapidoth_2-1" class="reference"><a href="#cite_note-Lapidoth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 464">: p. 464 </span></sup><sup id="cite_ref-Gubner_3-1" class="reference"><a href="#cite_note-Gubner-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.335">: p.335 </span></sup>
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 {K} _{\mathbf {X} \mathbf {X} }=\operatorname {Var} [\mathbf {X} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {X} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mi>Var</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {Var} [\mathbf {X} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {X} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}</annotation>
</semantics>
</math></span><img src="./832c09d3bfed857c48003e7fc636541db3d12f5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.432ex; height:3.176ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {Var} [\mathbf {X} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {X} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.3" class="reference nourlexpansion" style="font-weight:bold;">Eq.3</span></td></tr></tbody></table>
</div>
<p>By extension, the <b><a href="Cross-covariance_matrix" title="Cross-covariance matrix">cross-covariance matrix</a></b> between two random vectors <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> (<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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> having <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> elements and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> having <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 p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> elements) is the <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 n\times p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation>
</semantics>
</math></span><img src="./43ad58cdd60e9b0ab2bec828151c740accf92028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}" loading="lazy"></span> matrix<sup id="cite_ref-Gubner_3-2" class="reference"><a href="#cite_note-Gubner-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.336">: p.336 </span></sup>
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo stretchy="false">]</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}</annotation>
</semantics>
</math></span><img src="./2fc40423a3bffc9a6cbfef4423ca44b501b807ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:74.736ex; height:3.176ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{T}]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.4" class="reference nourlexpansion" style="font-weight:bold;">Eq.4</span></td></tr></tbody></table>
</div>
<p>where again the matrix expectation is taken element-by-element in the matrix. Here the (<i>i,j</i>)<sup>th</sup> element is the covariance between the <i>i</i><sup> th</sup> element of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and the <i>j</i><sup> th</sup> element of <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3></div>
<p>The covariance matrix is a <a href="Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>, i.e.<sup id="cite_ref-Lapidoth_2-2" class="reference"><a href="#cite_note-Lapidoth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 466">: p. 466 </span></sup>
</p>
<dl><dd><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 {K} _{\mathbf {X} \mathbf {X} }^{T}=\operatorname {K} _{\mathbf {X} \mathbf {X} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{T}=\operatorname {K} _{\mathbf {X} \mathbf {X} }}</annotation>
</semantics>
</math></span><img src="./2817b3e701e9ecedd571854018e34a948919b390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.891ex; height:3.176ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{T}=\operatorname {K} _{\mathbf {X} \mathbf {X} }}" loading="lazy"></span>.</dd></dl>
<p>The covariance matrix is a <a href="Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">positive semidefinite matrix</a>, i.e.<sup id="cite_ref-Lapidoth_2-3" class="reference"><a href="#cite_note-Lapidoth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 465">: p. 465 </span></sup>
</p>
<dl><dd><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 \mathbf {a} ^{T}\operatorname {K} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">a</mi>
</mrow>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
<mspace width="1em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>for all&nbsp;</mtext>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">a</mi>
</mrow>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ^{T}\operatorname {K} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./ac975ed48c12273147ee6d2a518d811f8d515643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.682ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} ^{T}\operatorname {K} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}" loading="lazy"></span>.</dd></dl>
<p>The cross-covariance matrix <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 {Cov} [\mathbf {Y} ,\mathbf {X} ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} [\mathbf {Y} ,\mathbf {X} ]}</annotation>
</semantics>
</math></span><img src="./e86490c56b48869d56250f25ecdb101769cff335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cov} [\mathbf {Y} ,\mathbf {X} ]}" loading="lazy"></span> is simply the transpose of the matrix <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 {Cov} [\mathbf {X} ,\mathbf {Y} ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]}</annotation>
</semantics>
</math></span><img src="./4d84d647f6dffb3803750a8f44ee3068f70f44f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]}" loading="lazy"></span>, i.e.
</p>
<dl><dd><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 {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }^{T}}</annotation>
</semantics>
</math></span><img src="./6c17d1bb7550e41dcc3a23f4ac503708186b43f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.891ex; height:3.176ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }^{T}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Uncorrelatedness">Uncorrelatedness</h3></div>
<p>Two random vectors <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 \mathbf {X} =(X_{1},...,X_{m})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},...,X_{m})^{T}}</annotation>
</semantics>
</math></span><img src="./a3e764681e36e20e77e63d7a8d7e8dd0f0aae6b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.064ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},...,X_{m})^{T}}" loading="lazy"></span> and <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 \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}</annotation>
</semantics>
</math></span><img src="./fc09536f2d119eb29acd3ff040e673be576f016a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.46ex; height:3.176ex;" alt="{\displaystyle \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}" loading="lazy"></span> are called <b>uncorrelated</b> if
</p>
<dl><dd><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} [\mathbf {X} \mathbf {Y} ^{T}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}</annotation>
</semantics>
</math></span><img src="./a42871987cbf77a6e3ecf8ecbb23e4f33912ebcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.972ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}" loading="lazy"></span>.</dd></dl>
<p>They are uncorrelated <a href="If_and_only_if" title="If and only if">if and only if</a> their cross-covariance matrix <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 {K} _{\mathbf {X} \mathbf {Y} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}</annotation>
</semantics>
</math></span><img src="./6a6d7b5411bd0a1b87e093e509a87efaed04d2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.896ex; height:2.509ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}" loading="lazy"></span> is zero.<sup id="cite_ref-Gubner_3-3" class="reference"><a href="#cite_note-Gubner-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.337">: p.337 </span></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Correlation_and_cross-correlation">Correlation and cross-correlation</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Definitions_2">Definitions</h3></div>
<p>The <b><a href="Autocorrelation_matrix" class="mw-redirect" title="Autocorrelation matrix">correlation matrix</a></b> (also called <b>second moment</b>) of an <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 n\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation>
</semantics>
</math></span><img src="./d24148f103e1cccb60addeeb0a64cb1c3d5622e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}" loading="lazy"></span> random vector is an <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> matrix whose (<i>i,j</i>)<sup>th</sup> element is the correlation between the <i>i</i><sup> th</sup> and the <i>j</i><sup> th</sup> random variables. The correlation matrix is the expected value, element by element, of the <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 n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> matrix computed as <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 \mathbf {X} \mathbf {X} ^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \mathbf {X} ^{T}}</annotation>
</semantics>
</math></span><img src="./632383f6ef09155f4b1c9f8193455ab77860ef4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.428ex; height:2.676ex;" alt="{\displaystyle \mathbf {X} \mathbf {X} ^{T}}" loading="lazy"></span>, where the superscript T refers to the transpose of the indicated vector:<sup id="cite_ref-Papoulis_4-0" class="reference"><a href="#cite_note-Papoulis-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.190">: p.190 </span></sup><sup id="cite_ref-Gubner_3-4" class="reference"><a href="#cite_note-Gubner-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.334">: p.334 </span></sup>
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 {R} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [\mathbf {X} \mathbf {X} ^{\mathrm {T} }]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">T</mi>
</mrow>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [\mathbf {X} \mathbf {X} ^{\mathrm {T} }]}</annotation>
</semantics>
</math></span><img src="./a4bb7af4f585dfa9c3e53939ad69802f1e112150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.232ex; height:3.176ex;" alt="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [\mathbf {X} \mathbf {X} ^{\mathrm {T} }]}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.5" class="reference nourlexpansion" style="font-weight:bold;">Eq.5</span></td></tr></tbody></table>
</div>
<p>By extension, the <b>cross-correlation matrix</b> between two random vectors <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> (<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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> having <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> elements and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> having <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 p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> elements) is the <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 n\times p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation>
</semantics>
</math></span><img src="./43ad58cdd60e9b0ab2bec828151c740accf92028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}" loading="lazy"></span> matrix
</p>
<div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]}</annotation>
</semantics>
</math></span><img src="./2068dba590bf3230ca69509a143ee7ce4013e979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.202ex; height:3.176ex;" alt="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_Eq.6" class="reference nourlexpansion" style="font-weight:bold;">Eq.6</span></td></tr></tbody></table>
</div>
<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>
<p>The correlation matrix is related to the covariance matrix by
</p>
<dl><dd><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 {R} _{\mathbf {X} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}</annotation>
</semantics>
</math></span><img src="./adb44f1d86d971aec8db66f846cfef11d848912a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.944ex; height:3.176ex;" alt="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}" loading="lazy"></span>.</dd></dl>
<p>Similarly for the cross-correlation matrix and the cross-covariance matrix:
</p>
<dl><dd><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 {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi mathvariant="normal">K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}</annotation>
</semantics>
</math></span><img src="./4efa28447cc9ee6d8b3b0023ab2fc652c5717f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.944ex; height:3.176ex;" alt="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }=\operatorname {K} _{\mathbf {X} \mathbf {Y} }+\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Orthogonality">Orthogonality</h2></div>
<p>Two random vectors of the same size <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 \mathbf {X} =(X_{1},...,X_{n})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}</annotation>
</semantics>
</math></span><img src="./bc319ec653bf15480e36b48c0d4029b8244dfbce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.607ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}" loading="lazy"></span> and <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 \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}</annotation>
</semantics>
</math></span><img src="./fc09536f2d119eb29acd3ff040e673be576f016a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.46ex; height:3.176ex;" alt="{\displaystyle \mathbf {Y} =(Y_{1},...,Y_{n})^{T}}" loading="lazy"></span> are called <b>orthogonal</b> if
</p>
<dl><dd><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} [\mathbf {X} ^{T}\mathbf {Y} ]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {X} ^{T}\mathbf {Y} ]=0}</annotation>
</semantics>
</math></span><img src="./b85a4e87960f920fc2b2e0d73fbc45ae39d9e0f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.566ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [\mathbf {X} ^{T}\mathbf {Y} ]=0}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Independence">Independence</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence (probability theory)</a></div>
<p>Two random vectors <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> are called <b>independent</b> if for all <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 \mathbf {x} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
</semantics>
</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> and <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 \mathbf {y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>
</semantics>
</math></span><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"></span>
</p>
<dl><dd><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 F_{\mathbf {X,Y} }(\mathbf {x,y} )=F_{\mathbf {X} }(\mathbf {x} )\cdot F_{\mathbf {Y} }(\mathbf {y} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
<mo mathvariant="bold">,</mo>
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
<mo mathvariant="bold">,</mo>
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )=F_{\mathbf {X} }(\mathbf {x} )\cdot F_{\mathbf {Y} }(\mathbf {y} )}</annotation>
</semantics>
</math></span><img src="./fd8491ad5d5c859ad662b18465bd258f0090c4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.396ex; height:3.009ex;" alt="{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )=F_{\mathbf {X} }(\mathbf {x} )\cdot F_{\mathbf {Y} }(\mathbf {y} )}" loading="lazy"></span></dd></dl>
<p>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 F_{\mathbf {X} }(\mathbf {x} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }(\mathbf {x} )}</annotation>
</semantics>
</math></span><img src="./e6138dae17f281ce237e6a47d8689f04fbb7fb15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.375ex; height:2.843ex;" alt="{\displaystyle F_{\mathbf {X} }(\mathbf {x} )}" loading="lazy"></span> and <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 F_{\mathbf {Y} }(\mathbf {y} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {Y} }(\mathbf {y} )}</annotation>
</semantics>
</math></span><img src="./eb3791b1f3525dcffceeee69c21f10d6e1681d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.375ex; height:2.843ex;" alt="{\displaystyle F_{\mathbf {Y} }(\mathbf {y} )}" loading="lazy"></span> denote the cumulative distribution functions of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> and<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 F_{\mathbf {X,Y} }(\mathbf {x,y} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
<mo mathvariant="bold">,</mo>
<mi mathvariant="bold">Y</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
<mo mathvariant="bold">,</mo>
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )}</annotation>
</semantics>
</math></span><img src="./d5a8450e06bca2a1744afd6ea28bdb93e131f2d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.868ex; height:3.009ex;" alt="{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )}" loading="lazy"></span> denotes their joint cumulative distribution function. Independence of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> is often denoted by <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 \mathbf {X} \perp \!\!\!\perp \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>⊥<!-- ⊥ --></mo>
<mspace width="negativethinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
<mo>⊥<!-- ⊥ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \perp \!\!\!\perp \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c9b1d7e8c33ad3e2584c07811ad34e57f252b82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.784ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} \perp \!\!\!\perp \mathbf {Y} }" loading="lazy"></span>.
Written component-wise, <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 \mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"></span> are called independent if for all <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 x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}}</annotation>
</semantics>
</math></span><img src="./7e80b7eba890f95783d131e68c68c6e2957beb39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.33ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}}" loading="lazy"></span>
</p>
<dl><dd><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 F_{X_{1},\ldots ,X_{m},Y_{1},\ldots ,Y_{n}}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})=F_{X_{1},\ldots ,X_{m}}(x_{1},\ldots ,x_{m})\cdot F_{Y_{1},\ldots ,Y_{n}}(y_{1},\ldots ,y_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{X_{1},\ldots ,X_{m},Y_{1},\ldots ,Y_{n}}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})=F_{X_{1},\ldots ,X_{m}}(x_{1},\ldots ,x_{m})\cdot F_{Y_{1},\ldots ,Y_{n}}(y_{1},\ldots ,y_{n})}</annotation>
</semantics>
</math></span><img src="./cd6b37390f3ed2f0925e8ad928fc3554ce4692e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:86.021ex; height:3.009ex;" alt="{\displaystyle F_{X_{1},\ldots ,X_{m},Y_{1},\ldots ,Y_{n}}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})=F_{X_{1},\ldots ,X_{m}}(x_{1},\ldots ,x_{m})\cdot F_{Y_{1},\ldots ,Y_{n}}(y_{1},\ldots ,y_{n})}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Characteristic_function">Characteristic function</h2></div>
<p>The <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of a random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> with <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> components is a function <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 \mathbb {R} ^{n}\to \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./a7b23f80a7cdfa734320f8c7a4afc7dc664b5b8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }" loading="lazy"></span> that maps every vector <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 \mathbf {\omega } =(\omega _{1},\ldots ,\omega _{n})^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {\omega } =(\omega _{1},\ldots ,\omega _{n})^{T}}</annotation>
</semantics>
</math></span><img src="./c4d3a2cbca0fa31fc4b2ec1e1ad0a7c1de114a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.085ex; height:3.176ex;" alt="{\displaystyle \mathbf {\omega } =(\omega _{1},\ldots ,\omega _{n})^{T}}" loading="lazy"></span> to a complex number. It is defined by<sup id="cite_ref-Lapidoth_2-4" class="reference"><a href="#cite_note-Lapidoth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 468">: p. 468 </span></sup>
</p>
<dl><dd><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 \varphi _{\mathbf {X} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i(\mathbf {\omega } ^{T}\mathbf {X} )}\right]=\operatorname {E} \left[e^{i(\omega _{1}X_{1}+\ldots +\omega _{n}X_{n})}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
</mrow>
<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>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi _{\mathbf {X} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i(\mathbf {\omega } ^{T}\mathbf {X} )}\right]=\operatorname {E} \left[e^{i(\omega _{1}X_{1}+\ldots +\omega _{n}X_{n})}\right]}</annotation>
</semantics>
</math></span><img src="./8c60146238888d3530b7411e867111a572489abc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.908ex; height:4.843ex;" alt="{\displaystyle \varphi _{\mathbf {X} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i(\mathbf {\omega } ^{T}\mathbf {X} )}\right]=\operatorname {E} \left[e^{i(\omega _{1}X_{1}+\ldots +\omega _{n}X_{n})}\right]}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Further_properties">Further properties</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Expectation_of_a_quadratic_form">Expectation of a quadratic form</h3></div>
<p>One can take the expectation of a <a href="Quadratic_form_(statistics)" title="Quadratic form (statistics)">quadratic form</a> in the random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> as follows:<sup id="cite_ref-Kendrick_5-0" class="reference"><a href="#cite_note-Kendrick-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.170–171">: p.170–171 </span></sup>
</p>
<dl><dd><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} [\mathbf {X} ^{T}A\mathbf {X} ]=\operatorname {E} [\mathbf {X} ]^{T}A\operatorname {E} [\mathbf {X} ]+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {X} ^{T}A\mathbf {X} ]=\operatorname {E} [\mathbf {X} ]^{T}A\operatorname {E} [\mathbf {X} ]+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }),}</annotation>
</semantics>
</math></span><img src="./3b9dfcc4b9bca76f1a66522e7b7603bd7adca96a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.375ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [\mathbf {X} ^{T}A\mathbf {X} ]=\operatorname {E} [\mathbf {X} ]^{T}A\operatorname {E} [\mathbf {X} ]+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }),}" loading="lazy"></span></dd></dl>
<p>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 K_{\mathbf {X} \mathbf {X} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{\mathbf {X} \mathbf {X} }}</annotation>
</semantics>
</math></span><img src="./4391acafd236f2534909b966fc1a1feb4033c6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.061ex; height:2.509ex;" alt="{\displaystyle K_{\mathbf {X} \mathbf {X} }}" loading="lazy"></span> is the covariance matrix of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and <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 {tr} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tr</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} }</annotation>
</semantics>
</math></span><img src="./7e779faa4557258ea25e8101f307317fc771a1ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \operatorname {tr} }" loading="lazy"></span> refers to the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of a matrix — that is, to the sum of the elements on its <a href="Main_diagonal" title="Main diagonal">main diagonal</a> (from upper left to lower right). Since the quadratic form is a scalar, so is its expectation.
</p><p><b>Proof</b>: Let <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 \mathbf {z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation>
</semantics>
</math></span><img src="./82eca5d0928078d5a61b9e7e98cc73db31070909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" loading="lazy"></span> be an <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 m\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times 1}</annotation>
</semantics>
</math></span><img src="./7f7ea91d81567531f5ef6d3b669be211ff953e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle m\times 1}" loading="lazy"></span> random vector with <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} [\mathbf {z} ]=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {z} ]=\mu }</annotation>
</semantics>
</math></span><img src="./a2c9a1f25ffb3b5b2239af348cc56c5b7ef47965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.565ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\mathbf {z} ]=\mu }" loading="lazy"></span> and <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 {Cov} [\mathbf {z} ]=V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} [\mathbf {z} ]=V}</annotation>
</semantics>
</math></span><img src="./7cc12db2602c6710843ecd22e6a1918f7ee86a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cov} [\mathbf {z} ]=V}" loading="lazy"></span> and let <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> be an <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 m\times m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times m}</annotation>
</semantics>
</math></span><img src="./367523981d714dcd9214703d654bfdedbe58d44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.921ex; height:1.676ex;" alt="{\displaystyle m\times m}" loading="lazy"></span> non-stochastic matrix.
</p><p>Then based on the formula for the covariance, if we denote <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 \mathbf {z} ^{T}=\mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} ^{T}=\mathbf {X} }</annotation>
</semantics>
</math></span><img src="./002c74c01566641419cf14756a890d55e97ee489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.695ex; height:2.676ex;" alt="{\displaystyle \mathbf {z} ^{T}=\mathbf {X} }" loading="lazy"></span> and <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 \mathbf {z} ^{T}A^{T}=\mathbf {Y} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} ^{T}A^{T}=\mathbf {Y} }</annotation>
</semantics>
</math></span><img src="./921906735cd4af53aa943094bff6b250c4e564a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.828ex; height:2.676ex;" alt="{\displaystyle \mathbf {z} ^{T}A^{T}=\mathbf {Y} }" loading="lazy"></span>, we see that:
</p>
<dl><dd><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 {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Y</mi>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}</annotation>
</semantics>
</math></span><img src="./e9d84e6ce8bc9899e40bf71a837a358dd49d9f17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.247ex; height:3.176ex;" alt="{\displaystyle \operatorname {Cov} [\mathbf {X} ,\mathbf {Y} ]=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}" loading="lazy"></span></dd></dl>
<p>Hence
</p>
<dl><dd><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 {\begin{aligned}\operatorname {E} [XY^{T}]&amp;=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}A\mu ,\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mi>z</mi>
<mo stretchy="false">]</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>+</mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>+</mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [XY^{T}]&amp;=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}A\mu ,\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./1248dedb225ea14b29d71dc4c7d97a07cbee7026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.775ex; margin-bottom: -0.229ex; width:44.986ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [XY^{T}]&amp;=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&amp;=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}A\mu ,\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>which leaves us to show that
</p>
<dl><dd><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 {Cov} [z^{T},z^{T}A^{T}]=\operatorname {tr} (AV).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>V</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} [z^{T},z^{T}A^{T}]=\operatorname {tr} (AV).}</annotation>
</semantics>
</math></span><img src="./7abae2aaa5d16ade185de7982d6debf5f8d5327b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.388ex; height:3.176ex;" alt="{\displaystyle \operatorname {Cov} [z^{T},z^{T}A^{T}]=\operatorname {tr} (AV).}" loading="lazy"></span></dd></dl>
<p>This is true based on the fact that one can <a href="Trace_(linear_algebra)#Properties" title="Trace (linear algebra)">cyclically permute matrices when taking a trace</a> without changing the end result (e.g.: <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 {tr} (AB)=\operatorname {tr} (BA)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}</annotation>
</semantics>
</math></span><img src="./95e3cc6cb1fb477706acf1c248d6f66882df7176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.364ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}" loading="lazy"></span>).
</p><p>We see <a href="Covariance#Definition" title="Covariance">that</a>
</p>
<dl><dd><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 {\begin{aligned}\operatorname {Cov} [z^{T},z^{T}A^{T}]&amp;=\operatorname {E} \left[\left(z^{T}-E(z^{T})\right)\left(z^{T}A^{T}-E\left(z^{T}A^{T}\right)\right)^{T}\right]\\&amp;=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&amp;=\operatorname {E} \left[(z-\mu )^{T}(Az-A\mu )\right].\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>E</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Cov} [z^{T},z^{T}A^{T}]&amp;=\operatorname {E} \left[\left(z^{T}-E(z^{T})\right)\left(z^{T}A^{T}-E\left(z^{T}A^{T}\right)\right)^{T}\right]\\&amp;=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&amp;=\operatorname {E} \left[(z-\mu )^{T}(Az-A\mu )\right].\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./bc6d588e480ee364b9a02a3ab231217bc8ddbfa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.094ex; margin-bottom: -0.244ex; width:57.892ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Cov} [z^{T},z^{T}A^{T}]&amp;=\operatorname {E} \left[\left(z^{T}-E(z^{T})\right)\left(z^{T}A^{T}-E\left(z^{T}A^{T}\right)\right)^{T}\right]\\&amp;=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&amp;=\operatorname {E} \left[(z-\mu )^{T}(Az-A\mu )\right].\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>And since
</p>
<dl><dd><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 \left({z-\mu }\right)^{T}\left({Az-A\mu }\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left({z-\mu }\right)^{T}\left({Az-A\mu }\right)}</annotation>
</semantics>
</math></span><img src="./6ce7255c2b558199d78c3695294acea5d9fae73a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.542ex; height:3.343ex;" alt="{\displaystyle \left({z-\mu }\right)^{T}\left({Az-A\mu }\right)}" loading="lazy"></span></dd></dl>
<p>is a <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, then
</p>
<dl><dd><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 (z-\mu )^{T}(Az-A\mu )=\operatorname {tr} \left({(z-\mu )^{T}(Az-A\mu )}\right)=\operatorname {tr} \left((z-\mu )^{T}A(z-\mu )\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (z-\mu )^{T}(Az-A\mu )=\operatorname {tr} \left({(z-\mu )^{T}(Az-A\mu )}\right)=\operatorname {tr} \left((z-\mu )^{T}A(z-\mu )\right)}</annotation>
</semantics>
</math></span><img src="./2e6f9b51cd233331ffa7bc8740f60edffd1cfd85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:69.809ex; height:3.343ex;" alt="{\displaystyle (z-\mu )^{T}(Az-A\mu )=\operatorname {tr} \left({(z-\mu )^{T}(Az-A\mu )}\right)=\operatorname {tr} \left((z-\mu )^{T}A(z-\mu )\right)}" loading="lazy"></span></dd></dl>
<p>trivially. Using the permutation we get:
</p>
<dl><dd><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 {tr} \left({(z-\mu )^{T}A(z-\mu )}\right)=\operatorname {tr} \left({A(z-\mu )(z-\mu )^{T}}\right),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left({(z-\mu )^{T}A(z-\mu )}\right)=\operatorname {tr} \left({A(z-\mu )(z-\mu )^{T}}\right),}</annotation>
</semantics>
</math></span><img src="./6f5ba98fe2cc12d785b9e8cbf3194d972417814e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.46ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left({(z-\mu )^{T}A(z-\mu )}\right)=\operatorname {tr} \left({A(z-\mu )(z-\mu )^{T}}\right),}" loading="lazy"></span></dd></dl>
<p>and by plugging this into the original formula we get:
</p>
<dl><dd><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 {\begin{aligned}\operatorname {Cov} \left[{z^{T},z^{T}A^{T}}\right]&amp;=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&amp;=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&amp;=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&amp;=\operatorname {tr} (AV).\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>E</mi>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>E</mi>
<mrow>
<mo>[</mo>
<mrow>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mi>V</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Cov} \left[{z^{T},z^{T}A^{T}}\right]&amp;=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&amp;=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&amp;=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&amp;=\operatorname {tr} (AV).\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./577d2617a04eb2dce0255889dbe7af3997cfabb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.663ex; margin-bottom: -0.175ex; width:45.732ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Cov} \left[{z^{T},z^{T}A^{T}}\right]&amp;=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&amp;=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&amp;=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&amp;=\operatorname {tr} (AV).\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Expectation_of_the_product_of_two_different_quadratic_forms">Expectation of the product of two different quadratic forms</h3></div>
<p>One can take the expectation of the product of two different quadratic forms in a zero-mean <a href="Joint_normality" class="mw-redirect" title="Joint normality">Gaussian</a> random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> as follows:<sup id="cite_ref-Kendrick_5-1" class="reference"><a href="#cite_note-Kendrick-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: pp. 162–176">: pp. 162–176 </span></sup>
</p>
<dl><dd><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} \left[(\mathbf {X} ^{T}A\mathbf {X} )(\mathbf {X} ^{T}B\mathbf {X} )\right]=2\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }BK_{\mathbf {X} \mathbf {X} })+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} })\operatorname {tr} (BK_{\mathbf {X} \mathbf {X} })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mi>B</mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>tr</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>B</mi>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[(\mathbf {X} ^{T}A\mathbf {X} )(\mathbf {X} ^{T}B\mathbf {X} )\right]=2\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }BK_{\mathbf {X} \mathbf {X} })+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} })\operatorname {tr} (BK_{\mathbf {X} \mathbf {X} })}</annotation>
</semantics>
</math></span><img src="./82326436dbf5e99dbe52e69dbddceb9b4150a9fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.516ex; height:3.343ex;" alt="{\displaystyle \operatorname {E} \left[(\mathbf {X} ^{T}A\mathbf {X} )(\mathbf {X} ^{T}B\mathbf {X} )\right]=2\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} }BK_{\mathbf {X} \mathbf {X} })+\operatorname {tr} (AK_{\mathbf {X} \mathbf {X} })\operatorname {tr} (BK_{\mathbf {X} \mathbf {X} })}" loading="lazy"></span></dd></dl>
<p>where again <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 K_{\mathbf {X} \mathbf {X} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{\mathbf {X} \mathbf {X} }}</annotation>
</semantics>
</math></span><img src="./4391acafd236f2534909b966fc1a1feb4033c6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.061ex; height:2.509ex;" alt="{\displaystyle K_{\mathbf {X} \mathbf {X} }}" loading="lazy"></span> is the covariance matrix of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span>. Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.
</p>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Portfolio_theory">Portfolio theory</h3></div>
<p>In <a href="Portfolio_theory" class="mw-redirect" title="Portfolio theory">portfolio theory</a> in <a href="Finance" title="Finance">finance</a>, an objective often is to choose a portfolio of risky assets such that the distribution of the random portfolio return has desirable properties. For example, one might want to choose the portfolio return having the lowest variance for a given expected value. Here the random vector is the vector <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 \mathbf {r} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>
</semantics>
</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span> of random returns on the individual assets, and the portfolio return <i>p</i> (a random scalar) is the inner product of the vector of random returns with a vector <i>w</i> of portfolio weights — the fractions of the portfolio placed in the respective assets. Since <i>p</i> = <i>w</i><sup>T</sup><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 \mathbf {r} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>
</semantics>
</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span>, the expected value of the portfolio return is <i>w</i><sup>T</sup>E(<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 \mathbf {r} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>
</semantics>
</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span>) and the variance of the portfolio return can be shown to be <i>w</i><sup>T</sup>C<i>w</i>, where C is the covariance matrix of <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 \mathbf {r} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>
</semantics>
</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Regression_theory">Regression theory</h3></div>
<p>In <a href="Linear_regression" title="Linear regression">linear regression</a> theory, we have data on <i>n</i> observations on a dependent variable <i>y</i> and <i>n</i> observations on each of <i>k</i> independent variables <i>x<sub>j</sub></i>. The observations on the dependent variable are stacked into a column vector <i>y</i>; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a <a href="Design_matrix" title="Design matrix">design matrix</a> <i>X</i> (not denoting a random vector in this context) of observations on the independent variables. Then the following regression equation is postulated as a description of the process that generated the data:
</p>
<dl><dd><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 y=X\beta +e,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>X</mi>
<mi>β<!-- β --></mi>
<mo>+</mo>
<mi>e</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=X\beta +e,}</annotation>
</semantics>
</math></span><img src="./e3940aae84e376feaf1abb80f096dc5bd866b157.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.137ex; height:2.509ex;" alt="{\displaystyle y=X\beta +e,}" loading="lazy"></span></dd></dl>
<p>where β is a postulated fixed but unknown vector of <i>k</i> response coefficients, and <i>e</i> is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such as <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>, a vector <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 {\hat {\beta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}}</annotation>
</semantics>
</math></span><img src="./efdb50e00928e4013750a476dab75eeb3cbd5799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.451ex; height:3.176ex;" alt="{\displaystyle {\hat {\beta }}}" loading="lazy"></span> is chosen as an estimate of β, and the estimate of the vector <i>e</i>, denoted <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 {\hat {e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}}</annotation>
</semantics>
</math></span><img src="./7ac09328845eecc01a117acbf303c1bc1decc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {e}}}" loading="lazy"></span>, is computed as
</p>
<dl><dd><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 {\hat {e}}=y-X{\hat {\beta }}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}=y-X{\hat {\beta }}.}</annotation>
</semantics>
</math></span><img src="./2e018ea3b7ebf9e90d53fd573d51123d9ea260bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.464ex; height:3.176ex;" alt="{\displaystyle {\hat {e}}=y-X{\hat {\beta }}.}" loading="lazy"></span></dd></dl>
<p>Then the statistician must analyze the properties of <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 {\hat {\beta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}}</annotation>
</semantics>
</math></span><img src="./efdb50e00928e4013750a476dab75eeb3cbd5799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.451ex; height:3.176ex;" alt="{\displaystyle {\hat {\beta }}}" loading="lazy"></span> and <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 {\hat {e}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>e</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}}</annotation>
</semantics>
</math></span><img src="./7ac09328845eecc01a117acbf303c1bc1decc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {e}}}" loading="lazy"></span>, which are viewed as random vectors since a randomly different selection of <i>n</i> cases to observe would have resulted in different values for them.
</p>
<div class="mw-heading mw-heading3"><h3 id="Vector_time_series">Vector time series</h3></div>
<p>The evolution of a <i>k</i>×1 random vector <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> through time can be modelled as a <a href="Vector_autoregression" title="Vector autoregression">vector autoregression</a> (VAR) as follows:
</p>
<dl><dd><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 \mathbf {X} _{t}=c+A_{1}\mathbf {X} _{t-1}+A_{2}\mathbf {X} _{t-2}+\cdots +A_{p}\mathbf {X} _{t-p}+\mathbf {e} _{t},\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>c</mi>
<mo>+</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>p</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{t}=c+A_{1}\mathbf {X} _{t-1}+A_{2}\mathbf {X} _{t-2}+\cdots +A_{p}\mathbf {X} _{t-p}+\mathbf {e} _{t},\,}</annotation>
</semantics>
</math></span><img src="./6eb905df49bd5615febd034a92ff977876e8acf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.201ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} _{t}=c+A_{1}\mathbf {X} _{t-1}+A_{2}\mathbf {X} _{t-2}+\cdots +A_{p}\mathbf {X} _{t-p}+\mathbf {e} _{t},\,}" loading="lazy"></span></dd></dl>
<p>where the <i>i</i>-periods-back vector observation <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 \mathbf {X} _{t-i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{t-i}}</annotation>
</semantics>
</math></span><img src="./202d72353259b2fbe7c7954fd65183c03215388f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.691ex; height:2.509ex;" alt="{\displaystyle \mathbf {X} _{t-i}}" loading="lazy"></span> is called the <i>i</i>-th lag of <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 \mathbf {X} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>
</semantics>
</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span>, <i>c</i> is a <i>k</i>&nbsp;×&nbsp;1 vector of constants (<a href="Y-intercept" title="Y-intercept">intercepts</a>), <i>A<sub>i</sub></i> is a time-invariant <i>k</i>&nbsp;×&nbsp;<i>k</i> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> and <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 \mathbf {e} _{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{t}}</annotation>
</semantics>
</math></span><img src="./1ac82f273b520165a6f2fe002bfa3005a5d3a2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.051ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{t}}" loading="lazy"></span> is a <i>k</i>&nbsp;×&nbsp;1 random vector of <a href="Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">error</a> terms.
</p>
<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 class="mw-references-wrap"><ol class="references">
<li id="cite_note-Gallager-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gallager_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */


.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}


/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFGallager2013" class="citation book cs1">Gallager, Robert G. (2013). <i>Stochastic Processes Theory for Applications</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1-107-03975-9</bdi>.</cite></span>
</li>
<li id="cite_note-Lapidoth-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lapidoth_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lapidoth_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lapidoth_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Lapidoth_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Lapidoth_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFTaboga2017" class="citation book cs1">Taboga, Marco (2017). <i>Lectures on Probability Theory and Mathematical Statistics</i>. CreateSpace Independent Publishing Platform. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1981369195</bdi>.</cite></span>
</li>
<li id="cite_note-Gubner-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gubner_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gubner_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Gubner_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Gubner_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Gubner_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGubner2006" class="citation book cs1">Gubner, John A. (2006). <i>Probability and Random Processes for Electrical and Computer Engineers</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-521-86470-1</bdi>.</cite></span>
</li>
<li id="cite_note-Papoulis-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Papoulis_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFPapoulis1991" class="citation book cs1">Papoulis, Athanasius (1991). <i>Probability, Random Variables and Stochastic Processes</i> (Third&nbsp;ed.). McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-07-048477-5</bdi>.</cite></span>
</li>
<li id="cite_note-Kendrick-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kendrick_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kendrick_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFKendrick1981" class="citation book cs1">Kendrick, David (1981). <i>Stochastic Control for Economic Models</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-07-033962-7</bdi>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><cite id="CITEREFStarkWoods2012" class="citation book cs1">Stark, Henry; Woods, John W. (2012). "Random Vectors". <i>Probability, Statistics, and Random Processes for Engineers</i> (Fourth&nbsp;ed.). Pearson. pp.&nbsp;<span class="nowrap">295–</span>339. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-13-231123-6</bdi>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-07-30" href="https://en.wikipedia.org/wiki/?title=Multivariate_random_variable&amp;oldid=1303428131">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>