Multivariate random variable

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href="#Operations_on_random_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Operations on random vectors</span> </div> </a> <button aria-controls="toc-Operations_on_random_vectors-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Operations on random vectors subsection</span> </button> <ul id="toc-Operations_on_random_vectors-sublist" class="vector-toc-list"> <li id="toc-Affine_transformations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Affine transformations</span> </div> </a> <ul id="toc-Affine_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invertible_mappings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invertible_mappings"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Invertible mappings</span> </div> </a> <ul id="toc-Invertible_mappings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Expected_value" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expected_value"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Expected value</span> </div> </a> <ul id="toc-Expected_value-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covariance_and_cross-covariance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Covariance_and_cross-covariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Covariance and cross-covariance</span> </div> </a> <button aria-controls="toc-Covariance_and_cross-covariance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Covariance and cross-covariance subsection</span> </button> <ul id="toc-Covariance_and_cross-covariance-sublist" class="vector-toc-list"> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncorrelatedness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncorrelatedness"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Uncorrelatedness</span> </div> </a> <ul id="toc-Uncorrelatedness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Correlation_and_cross-correlation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Correlation_and_cross-correlation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Correlation and cross-correlation</span> </div> </a> <button aria-controls="toc-Correlation_and_cross-correlation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Correlation and cross-correlation subsection</span> </button> <ul id="toc-Correlation_and_cross-correlation-sublist" class="vector-toc-list"> <li id="toc-Definitions_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitions_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orthogonality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orthogonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Orthogonality</span> </div> </a> <ul id="toc-Orthogonality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Independence"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Independence</span> </div> </a> <ul id="toc-Independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characteristic_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Characteristic_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Characteristic function</span> </div> </a> <ul id="toc-Characteristic_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further properties</span> </div> </a> <button aria-controls="toc-Further_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further properties subsection</span> </button> <ul id="toc-Further_properties-sublist" class="vector-toc-list"> <li id="toc-Expectation_of_a_quadratic_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expectation_of_a_quadratic_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Expectation of a quadratic form</span> </div> </a> <ul id="toc-Expectation_of_a_quadratic_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expectation_of_the_product_of_two_different_quadratic_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expectation_of_the_product_of_two_different_quadratic_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Expectation of the product of two different quadratic forms</span> </div> </a> <ul id="toc-Expectation_of_the_product_of_two_different_quadratic_forms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Portfolio_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Portfolio_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Portfolio theory</span> </div> </a> <ul id="toc-Portfolio_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regression_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regression_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Regression theory</span> </div> </a> <ul id="toc-Regression_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_time_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_time_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Vector time series</span> </div> </a> <ul id="toc-Vector_time_series-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Multivariate random variable</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Vector_aleatori" title="Vector aleatori – Catalan" lang="ca" hreflang="ca" data-title="Vector aleatori" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C3%ADcerozm%C4%9Brn%C3%A1_n%C3%A1hodn%C3%A1_prom%C4%9Bnn%C3%A1" title="Vícerozměrná náhodná proměnná – Czech" lang="cs" hreflang="cs" data-title="Vícerozměrná náhodná proměnná" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zufallsvariable#Mehrdimensionale_Zufallsvariable" title="Zufallsvariable – German" lang="de" hreflang="de" data-title="Zufallsvariable" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector_aleatorio" title="Vector aleatorio – Spanish" lang="es" hreflang="es" data-title="Vector aleatorio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vecteur_al%C3%A9atoire" title="Vecteur aléatoire – French" lang="fr" hreflang="fr" data-title="Vecteur aléatoire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variabile_casuale_multivariata" title="Variabile casuale multivariata – Italian" lang="it" hreflang="it" data-title="Variabile casuale multivariata" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%9E%D7%A7%D7%A8%D7%99" title="וקטור מקרי – Hebrew" lang="he" hreflang="he" data-title="וקטור מקרי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gi_vektorv%C3%A1ltoz%C3%B3" title="Valószínűségi vektorváltozó – Hungarian" lang="hu" hreflang="hu" data-title="Valószínűségi vektorváltozó" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zmienna_losowa#Uogólnienia" title="Zmienna losowa – Polish" lang="pl" hreflang="pl" data-title="Zmienna losowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Многомерная случайная величина – Russian" lang="ru" hreflang="ru" data-title="Многомерная случайная величина" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D0%B3%D0%B0%D1%82%D0%BE%D0%B2%D0%B8%D0%BC%D1%96%D1%80%D0%BD%D0%B0_%D0%B2%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Багатовимірна випадкова величина – Ukrainian" lang="uk" hreflang="uk" data-title="Багатовимірна випадкова величина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q3179949#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div 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class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a 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statistics</a>.</div> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability</a>, and <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>multivariate random variable</b> or <b>random vector</b> is a list or <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vector</a> of mathematical <a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Random_variable" title="Random variable">random variables</a>, e.g. a <a href="/wiki/Random_matrix" title="Random matrix">random matrix</a>, <a href="/wiki/Random_tree" title="Random tree">random tree</a>, <a href="/wiki/Random_sequence" title="Random sequence">random sequence</a>, <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>, etc. </p><p>Formally, a multivariate random variable is a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada5a9bc21cd37cf25c8cfb2fd2408efa3cd09a6" 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}}}"></span> (or its <a href="/wiki/Transpose" title="Transpose">transpose</a>, which is a <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a>) whose components are <a href="/wiki/Random_variable" title="Random variable">random variables</a> on the <a href="/wiki/Probability_space" title="Probability space">probability space</a> <span class="mwe-math-element"><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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" 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)}"></span>, where <span class="mwe-math-element"><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</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 }"></span> is the <a href="/wiki/Sample_space" title="Sample space">sample space</a>, <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" 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}}}"></span> is the <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> (the collection of all events), and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"></span> is the <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> (a function returning each event's <a href="/wiki/Probability" title="Probability">probability</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Probability_distribution">Probability distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=1" title="Edit section: Probability distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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}}"></span> with the <a href="/wiki/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="/wiki/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="/wiki/Probability_distribution" title="Probability distribution">distributions</a> of each of the component random variables <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" 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}}"></span> are called <a href="/wiki/Marginal_distribution" title="Marginal distribution">marginal distributions</a>. The <a href="/wiki/Conditional_probability_distribution" title="Conditional probability distribution">conditional probability distribution</a> of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" 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}}"></span> given <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" 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}}"></span> is the probability distribution of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" 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}}"></span> when <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" 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}}"></span> is known to be a particular value. </p><p>The <b>cumulative distribution function</b> <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b30f84c300818779899583633faa44ef4cc9100" 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]}"></span> of a random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada5a9bc21cd37cf25c8cfb2fd2408efa3cd09a6" 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}}}"></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">.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}}</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ad8b4311ebf247c4854daeb4b4c76dbcbfc00e" 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})}"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/65bfe6b31f3dd0da617d72d142c1ece00a5b9093" 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}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Operations_on_random_vectors">Operations on random vectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=2" title="Edit section: Operations on random vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Random vectors can be subjected to the same kinds of <a href="/wiki/Euclidean_vector#Basic_properties" title="Euclidean vector">algebraic operations</a> as can non-random vectors: addition, subtraction, multiplication by a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, and the taking of <a href="/wiki/Dot_product" title="Dot product">inner products</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_transformations">Affine transformations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=3" title="Edit section: Affine transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Similarly, a new random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> can be defined by applying an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f147e7d7a52fb587d2eb46a4a767351ecfe6c5f" 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}}"></span> to a random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span>: </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/351eda808c74d82a5e4865c90c3210defa645b02" 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}"></span>, where <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" 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} }"></span> is an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" 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}"></span> matrix and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" 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}"></span> is an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24148f103e1cccb60addeeb0a64cb1c3d5622e0" 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}"></span> column vector.</dd></dl> <p>If <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" 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} }"></span> is an invertible matrix and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8c6feeb921a6d81808b98802b5caee9ba103a3" 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} }"></span> has a probability density function <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/51cce7149cdbfef66c1b70ccfcc069a5031e330c" 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} }}"></span>, then the probability density of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> is </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3306fba2ce1fd82b8ef6e95ee0946a678b221cd2" 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} |}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Invertible_mappings">Invertible mappings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=4" title="Edit section: Invertible mappings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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.290–291">: p.290–291 </span></sup> </p><p>Let <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"></span> be a one-to-one mapping from an open subset <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" 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}}}"></span> of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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}}"></span> onto a subset <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" 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}}}"></span> of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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}}"></span>, let <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"></span> have continuous partial derivatives in <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" 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}}}"></span> and let the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian determinant</a> of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"></span> be zero at no point of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" 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}}}"></span>. Assume that the real random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> has a probability density function <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdd0c4f093a3b1ad94514b03d24663825a311bc" 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} )}"></span> and satisfies <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a72fea4609d49280a2a257f7fc0ece78f6162e" 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}"></span>. Then the random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87935f4907978af41f9d3e2a3e7f9c64911b97c" 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} )}"></span> is of probability density </p> <dl><dd><span class="mwe-math-element"><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 {\frac {\partial \mathbf {x} }{\partial \mathbf {y} }}\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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> </mfrac> </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 {\frac {\partial \mathbf {x} }{\partial \mathbf {y} }}\right|}}\right|_{\mathbf {x} =g^{-1}(\mathbf {y} )}\mathbf {1} (\mathbf {y} \in R_{\mathbf {Y} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eece69ee5e3aa518d81d6914ce04bf34aa300d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:36.545ex; height:10.009ex;" alt="{\displaystyle \left.f_{\mathbf {Y} }(\mathbf {y} )={\frac {f_{\mathbf {X} }(\mathbf {x} )}{\left|\det {\frac {\partial \mathbf {x} }{\partial \mathbf {y} }}\right|}}\right|_{\mathbf {x} =g^{-1}(\mathbf {y} )}\mathbf {1} (\mathbf {y} \in R_{\mathbf {Y} })}"></span></dd></dl> <p>where <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" 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} }"></span> denotes the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> and set <span class="mwe-math-element"><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>></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} )>0\}\subseteq {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d2501f21cde86999c8614608b1eabd14ea13dc" 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}}}"></span> denotes support of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Expected_value">Expected value</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=5" title="Edit section: Expected value"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Expected_value" title="Expected value">expected value</a> or mean of a random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> is a fixed vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3027b4ba8627bb297a8039b230b2afb29a717422" 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} ]}"></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2b77d329025627999ff8a4ac19c3dcb12c8af1" 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} }}"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=6" title="Edit section: Covariance and cross-covariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=7" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24148f103e1cccb60addeeb0a64cb1c3d5622e0" 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}"></span> random vector is an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" 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}"></span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> whose (<i>i,j</i>)<sup>th</sup> element is the <a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" 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}"></span> matrix <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">computed as</a> <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/26928071db5f676c48f2cf3cdbd4bf86d06f052f" 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}}"></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/832c09d3bfed857c48003e7fc636541db3d12f5a" 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}}"></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="/wiki/Cross-covariance_matrix" title="Cross-covariance matrix">cross-covariance matrix</a></b> between two random vectors <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> (<span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> having <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> elements and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> having <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span> elements) is the <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" 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}"></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc40423a3bffc9a6cbfef4423ca44b501b807ab" 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}}"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and the <i>j</i><sup> th</sup> element of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=8" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The covariance matrix is a <a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/2817b3e701e9ecedd571854018e34a948919b390" 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} }}"></span>.</dd></dl> <p>The covariance matrix is a <a href="/wiki/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"><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" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac975ed48c12273147ee6d2a518d811f8d515643" 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}}"></span>.</dd></dl> <p>The cross-covariance matrix <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86490c56b48869d56250f25ecdb101769cff335" 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} ]}"></span> is simply the transpose of the matrix <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d84d647f6dffb3803750a8f44ee3068f70f44f7" 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} ]}"></span>, i.e. </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c17d1bb7550e41dcc3a23f4ac503708186b43f0" 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}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Uncorrelatedness">Uncorrelatedness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=9" title="Edit section: Uncorrelatedness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two random vectors <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e764681e36e20e77e63d7a8d7e8dd0f0aae6b4" 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}}"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc09536f2d119eb29acd3ff040e673be576f016a" 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}}"></span> are called <b>uncorrelated</b> if </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42871987cbf77a6e3ecf8ecbb23e4f33912ebcc" 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}}"></span>.</dd></dl> <p>They are uncorrelated if and only if their cross-covariance matrix <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6d7b5411bd0a1b87e093e509a87efaed04d2bd" 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} }}"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=10" title="Edit section: Correlation and cross-correlation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definitions_2">Definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=11" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24148f103e1cccb60addeeb0a64cb1c3d5622e0" 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}"></span> random vector is an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" 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}"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" 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}"></span> matrix computed as <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/632383f6ef09155f4b1c9f8193455ab77860ef4e" 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}}"></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4bb7af4f585dfa9c3e53939ad69802f1e112150" 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} }]}"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> (<span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> having <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> elements and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> having <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span> elements) is the <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" 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}"></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/2068dba590bf3230ca69509a143ee7ce4013e979" 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}]}"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=12" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The correlation matrix is related to the covariance matrix by </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/adb44f1d86d971aec8db66f846cfef11d848912a" 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}}"></span>.</dd></dl> <p>Similarly for the cross-correlation matrix and the cross-covariance matrix: </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/4efa28447cc9ee6d8b3b0023ab2fc652c5717f2e" 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}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Orthogonality">Orthogonality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=13" title="Edit section: Orthogonality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two random vectors of the same size <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc319ec653bf15480e36b48c0d4029b8244dfbce" 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}}"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc09536f2d119eb29acd3ff040e673be576f016a" 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}}"></span> are called <b>orthogonal</b> if </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/b85a4e87960f920fc2b2e0d73fbc45ae39d9e0f7" 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}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Independence">Independence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=14" title="Edit section: Independence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence (probability theory)</a></div> <p>Two random vectors <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> are called <b>independent</b> if for all <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f" 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} }"></span> </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8491ad5d5c859ad662b18465bd258f0090c4bb" 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} )}"></span></dd></dl> <p>where <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6138dae17f281ce237e6a47d8689f04fbb7fb15" 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} )}"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3791b1f3525dcffceeee69c21f10d6e1681d14" 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} )}"></span> denote the cumulative distribution functions of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> and<span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a8450e06bca2a1744afd6ea28bdb93e131f2d4" 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} )}"></span> denotes their joint cumulative distribution function. Independence of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> is often denoted by <span class="mwe-math-element"><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 width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b1d7e8c33ad3e2584c07811ad34e57f252b82e" 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} }"></span>. Written component-wise, <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92a7716a99fadda050469747fce1e475e0ec549" 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} }"></span> are called independent if for all <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e80b7eba890f95783d131e68c68c6e2957beb39" 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}}"></span> </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd6b37390f3ed2f0925e8ad928fc3554ce4692e8" 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})}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Characteristic_function">Characteristic function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=15" title="Edit section: Characteristic function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of a random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> with <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> components is a function <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b23f80a7cdfa734320f8c7a4afc7dc664b5b8f" 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} }"></span> that maps every vector <span class="mwe-math-element"><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</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d3a2cbca0fa31fc4b2ec1e1ad0a7c1de114a80" 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}}"></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"><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</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</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</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</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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c60146238888d3530b7411e867111a572489abc" 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]}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Further_properties">Further properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=16" title="Edit section: Further properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Expectation_of_a_quadratic_form">Expectation of a quadratic form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=17" title="Edit section: Expectation of a quadratic form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can take the expectation of a <a href="/wiki/Quadratic_form_(statistics)" title="Quadratic form (statistics)">quadratic form</a> in the random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9dfcc4b9bca76f1a66522e7b7603bd7adca96a" 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} }),}"></span></dd></dl> <p>where <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/4391acafd236f2534909b966fc1a1feb4033c6a7" 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} }}"></span> is the covariance matrix of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e779faa4557258ea25e8101f307317fc771a1ef" 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} }"></span> refers to the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of a matrix — that is, to the sum of the elements on its main diagonal (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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/82eca5d0928078d5a61b9e7e98cc73db31070909" 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} }"></span> be an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7ea91d81567531f5ef6d3b669be211ff953e6a" 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}"></span> random vector with <span class="mwe-math-element"><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</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\mathbf {z} ]=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c9a1f25ffb3b5b2239af348cc56c5b7ef47965" 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 }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc12db2602c6710843ecd22e6a1918f7ee86a07" 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}"></span> and let <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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}"></span> be an <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/367523981d714dcd9214703d654bfdedbe58d44a" 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}"></span> non-stochastic matrix. </p><p>Then based on the formula for the covariance, if we denote <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/002c74c01566641419cf14756a890d55e97ee489" 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} }"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/921906735cd4af53aa943094bff6b250c4e564a9" 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} }"></span>, we see that: </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d84e6ce8bc9899e40bf71a837a358dd49d9f17" 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}}"></span></dd></dl> <p>Hence </p> <dl><dd><span class="mwe-math-element"><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}]&=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&=\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> <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</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>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> <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</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>A</mi> <mi>μ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [XY^{T}]&=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}A\mu ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1248dedb225ea14b29d71dc4c7d97a07cbee7026" 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}]&=\operatorname {Cov} [X,Y]+\operatorname {E} [X]\operatorname {E} [Y]^{T}\\\operatorname {E} [z^{T}Az]&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\operatorname {E} [z^{T}]\operatorname {E} [z^{T}A^{T}]^{T}\\&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}(\mu ^{T}A^{T})^{T}\\&=\operatorname {Cov} [z^{T},z^{T}A^{T}]+\mu ^{T}A\mu ,\end{aligned}}}"></span></dd></dl> <p>which leaves us to show that </p> <dl><dd><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abae2aaa5d16ade185de7982d6debf5f8d5327b" 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).}"></span></dd></dl> <p>This is true based on the fact that one can <a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3cc6cb1fb477706acf1c248d6f66882df7176" 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)}"></span>). </p><p>We see <a href="/wiki/Covariance#Definition" title="Covariance">that</a> </p> <dl><dd><span class="mwe-math-element"><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}]&=\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]\\&=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&=\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> <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</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</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> <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</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</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}]&=\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]\\&=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&=\operatorname {E} \left[(z-\mu )^{T}(Az-A\mu )\right].\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6d588e480ee364b9a02a3ab231217bc8ddbfa6" 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}]&=\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]\\&=\operatorname {E} \left[(z^{T}-\mu ^{T})(z^{T}A^{T}-\mu ^{T}A^{T})^{T}\right]\\&=\operatorname {E} \left[(z-\mu )^{T}(Az-A\mu )\right].\end{aligned}}}"></span></dd></dl> <p>And since </p> <dl><dd><span class="mwe-math-element"><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</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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce7255c2b558199d78c3695294acea5d9fae73a" 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)}"></span></dd></dl> <p>is a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, then </p> <dl><dd><span class="mwe-math-element"><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</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</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</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</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</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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6f9b51cd233331ffa7bc8740f60edffd1cfd85" 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)}"></span></dd></dl> <p>trivially. Using the permutation we get: </p> <dl><dd><span class="mwe-math-element"><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</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</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</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5ba98fe2cc12d785b9e8cbf3194d972417814e" 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),}"></span></dd></dl> <p>and by plugging this into the original formula we get: </p> <dl><dd><span class="mwe-math-element"><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]&=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&=\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</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</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <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</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>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> <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</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>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> <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]&=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&=\operatorname {tr} (AV).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d2617a04eb2dce0255889dbe7af3997cfabb1" 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]&=E\left[{\left({z-\mu }\right)^{T}(Az-A\mu )}\right]\\&=E\left[\operatorname {tr} \left(A(z-\mu )(z-\mu )^{T}\right)\right]\\&=\operatorname {tr} \left({A\cdot \operatorname {E} \left((z-\mu )(z-\mu )^{T}\right)}\right)\\&=\operatorname {tr} (AV).\end{aligned}}}"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=18" title="Edit section: Expectation of the product of two different quadratic forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can take the expectation of the product of two different quadratic forms in a zero-mean <a href="/wiki/Joint_normality" class="mw-redirect" title="Joint normality">Gaussian</a> random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/82326436dbf5e99dbe52e69dbddceb9b4150a9fa" 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} })}"></span></dd></dl> <p>where again <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/4391acafd236f2534909b966fc1a1feb4033c6a7" 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} }}"></span> is the covariance matrix of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=19" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Portfolio_theory">Portfolio theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=20" title="Edit section: Portfolio theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Portfolio_theory" class="mw-redirect" title="Portfolio theory">portfolio theory</a> in <a href="/wiki/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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" 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} }"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" 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} }"></span>, the expected value of the portfolio return is <i>w</i><sup>T</sup>E(<span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" 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} }"></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"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" 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} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Regression_theory">Regression theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=21" title="Edit section: Regression theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/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="/wiki/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"><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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3940aae84e376feaf1abb80f096dc5bd866b157" 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,}"></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="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>, a vector <span class="mwe-math-element"><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</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdb50e00928e4013750a476dab75eeb3cbd5799" 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 }}}"></span> is chosen as an estimate of β, and the estimate of the vector <i>e</i>, denoted <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac09328845eecc01a117acbf303c1bc1decc4a1" 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}}}"></span>, is computed as </p> <dl><dd><span class="mwe-math-element"><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</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="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e018ea3b7ebf9e90d53fd573d51123d9ea260bd" 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 }}.}"></span></dd></dl> <p>Then the statistician must analyze the properties of <span class="mwe-math-element"><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</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdb50e00928e4013750a476dab75eeb3cbd5799" 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 }}}"></span> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac09328845eecc01a117acbf303c1bc1decc4a1" 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}}}"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=22" title="Edit section: Vector time series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The evolution of a <i>k</i>×1 random vector <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span> through time can be modelled as a <a href="/wiki/Vector_autoregression" title="Vector autoregression">vector autoregression</a> (VAR) as follows: </p> <dl><dd><span class="mwe-math-element"><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" /> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb905df49bd5615febd034a92ff977876e8acf4" 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},\,}"></span></dd></dl> <p>where the <i>i</i>-periods-back vector observation <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/202d72353259b2fbe7c7954fd65183c03215388f" 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}}"></span> is called the <i>i</i>-th lag of <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" 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} }"></span>, <i>c</i> is a <i>k</i> × 1 vector of constants (<a href="/wiki/Y-intercept" title="Y-intercept">intercepts</a>), <i>A<sub>i</sub></i> is a time-invariant <i>k</i> × <i>k</i> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> and <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac82f273b520165a6f2fe002bfa3005a5d3a2e8" 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}}"></span> is a <i>k</i> × 1 random vector of <a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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}</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">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFGallager2013" class="citation book cs1">Gallager, Robert G. (2013). <i>Stochastic Processes Theory for Applications</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-03975-9" title="Special:BookSources/978-1-107-03975-9"><bdi>978-1-107-03975-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Processes+Theory+for+Applications&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-107-03975-9&rft.aulast=Gallager&rft.aufirst=Robert+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLapidoth2009" class="citation book cs1">Lapidoth, Amos (2009). <i>A Foundation in Digital Communication</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19395-5" title="Special:BookSources/978-0-521-19395-5"><bdi>978-0-521-19395-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Foundation+in+Digital+Communication&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-19395-5&rft.aulast=Lapidoth&rft.aufirst=Amos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86470-1" title="Special:BookSources/978-0-521-86470-1"><bdi>978-0-521-86470-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Random+Processes+for+Electrical+and+Computer+Engineers&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-86470-1&rft.aulast=Gubner&rft.aufirst=John+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapoulis1991" class="citation book cs1">Papoulis, Athanasius (1991). <i>Probability, Random Variables and Stochastic Processes</i> (Third ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-048477-5" title="Special:BookSources/0-07-048477-5"><bdi>0-07-048477-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%2C+Random+Variables+and+Stochastic+Processes&rft.edition=Third&rft.pub=McGraw-Hill&rft.date=1991&rft.isbn=0-07-048477-5&rft.aulast=Papoulis&rft.aufirst=Athanasius&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKendrick1981" class="citation book cs1">Kendrick, David (1981). <i>Stochastic Control for Economic Models</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-033962-7" title="Special:BookSources/0-07-033962-7"><bdi>0-07-033962-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Control+for+Economic+Models&rft.pub=McGraw-Hill&rft.date=1981&rft.isbn=0-07-033962-7&rft.aulast=Kendrick&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_random_variable&action=edit&section=24" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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 ed.). Pearson. pp. <span class="nowrap">295–</span>339. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-231123-6" title="Special:BookSources/978-0-13-231123-6"><bdi>978-0-13-231123-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Random+Vectors&rft.btitle=Probability%2C+Statistics%2C+and+Random+Processes+for+Engineers&rft.pages=%3Cspan+class%3D%22nowrap%22%3E295-%3C%2Fspan%3E339&rft.edition=Fourth&rft.pub=Pearson&rft.date=2012&rft.isbn=978-0-13-231123-6&rft.aulast=Stark&rft.aufirst=Henry&rft.au=Woods%2C+John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+random+variable" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Multivariate_random_variable&oldid=1237686562">https://en.wikipedia.org/w/index.php?title=Multivariate_random_variable&oldid=1237686562</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Multivariate_statistics" title="Category:Multivariate statistics">Multivariate statistics</a></li><li><a href="/wiki/Category:Algebra_of_random_variables" title="Category:Algebra of random variables">Algebra of random variables</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 July 2024, at 01:02<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Multivariate random variable - Wikipedia