Mixture distribution

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href="#Uncountable_mixtures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Uncountable mixtures</span> </div> </a> <ul id="toc-Uncountable_mixtures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixtures_within_a_parametric_family" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mixtures_within_a_parametric_family"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mixtures within a parametric family</span> </div> </a> <ul id="toc-Mixtures_within_a_parametric_family-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet 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id="toc-A_normal_and_a_Cauchy_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_normal_and_a_Cauchy_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>A normal and a Cauchy distribution</span> </div> </a> <ul id="toc-A_normal_and_a_Cauchy_distribution-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">6</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <button 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searchaux" style="display:none">Probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mixture_model" title="Mixture model">Mixture model</a> and <a href="/wiki/Compound_probability_distribution" title="Compound probability distribution">Compound probability distribution</a></div> <p>In <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>mixture distribution</b> is the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">random vectors</a> (each having the same dimension), in which case the mixture distribution is a <a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">multivariate distribution</a>. </p><p>In cases where each of the underlying random variables is <a href="/wiki/Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous</a>, the outcome variable will also be continuous and its <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> is sometimes referred to as a <b>mixture density</b>. The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (and the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> if it exists) can be expressed as a <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a> (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions. The individual distributions that are combined to form the mixture distribution are called the <b>mixture components</b>, and the probabilities (or weights) associated with each component are called the <b>mixture weights</b>. The number of components in a mixture distribution is often restricted to being finite, although in some cases the components may be <a href="/wiki/Countable" class="mw-redirect" title="Countable">countably infinite</a> in number. More general cases (i.e. an <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> set of component distributions), as well as the countable case, are treated under the title of <b><a href="/wiki/Compound_probability_distribution" title="Compound probability distribution">compound distributions</a></b>. </p><p>A distinction needs to be made between a <a href="/wiki/Random_variable" title="Random variable">random variable</a> whose distribution function or density is the sum of a set of components (i.e. a mixture distribution) and a random variable whose value is the sum of the values of two or more underlying random variables, in which case the distribution is given by the <a href="/wiki/Convolution" title="Convolution">convolution</a> operator. As an example, the sum of two <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">jointly normally distributed</a> random variables, each with different means, will still have a normal distribution. On the other hand, a mixture density created as a mixture of two normal distributions with different means will have two peaks provided that the two means are far enough apart, showing that this distribution is radically different from a normal distribution. </p><p>Mixture distributions arise in many contexts in the literature and arise naturally where a <a href="/wiki/Statistical_population" title="Statistical population">statistical population</a> contains two or more <a href="/wiki/Subpopulation" class="mw-redirect" title="Subpopulation">subpopulations</a>. They are also sometimes used as a means of representing non-normal distributions. Data analysis concerning <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> involving mixture distributions is discussed under the title of <a href="/wiki/Mixture_model" title="Mixture model">mixture models</a>, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Finite_and_countable_mixtures">Finite and countable mixtures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=1" title="Edit section: Finite and countable mixtures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gaussian-mixture-example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Gaussian-mixture-example.svg/220px-Gaussian-mixture-example.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Gaussian-mixture-example.svg/330px-Gaussian-mixture-example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Gaussian-mixture-example.svg/440px-Gaussian-mixture-example.svg.png 2x" data-file-width="540" data-file-height="360" /></a><figcaption>Density of a mixture of three normal distributions (<i>μ</i> = 5, 10, 15, <i>σ</i> = 2) with equal weights. Each component is shown as a weighted density (each integrating to 1/3)</figcaption></figure> <p>Given a finite set of probability density functions <i>p</i><sub>1</sub>(<i>x</i>), ..., <i>p<sub>n</sub></i>(<i>x</i>), or corresponding cumulative distribution functions <i>P</i><sub>1</sub>(<i>x</i>), ..., <i>P<sub>n</sub></i>(<i>x</i>) and <b>weights</b> <i>w</i><sub>1</sub>, ..., <i>w<sub>n</sub></i> such that <span class="nowrap"><i>w<sub>i</sub></i> ≥ 0</span> and <span class="nowrap">Σ<i>w<sub>i</sub></i> = 1, </span> the mixture distribution can be represented by writing either the density, <i>f</i>, or the distribution function, <i>F</i>, as a sum (which in both cases is a convex combination): </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)=\sum _{i=1}^{n}\,w_{i}\,P_{i}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\sum _{i=1}^{n}\,w_{i}\,P_{i}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bd5bec18aa3e3978e36af007d32c3b304ffd5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.036ex; height:6.843ex;" alt="{\displaystyle F(x)=\sum _{i=1}^{n}\,w_{i}\,P_{i}(x),}"></span></dd> <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)=\sum _{i=1}^{n}\,w_{i}\,p_{i}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{i=1}^{n}\,w_{i}\,p_{i}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c56b7d1965b7fe50a1c4add03df6812d7a38f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.251ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{i=1}^{n}\,w_{i}\,p_{i}(x).}"></span></dd></dl> <p>This type of mixture, being a finite sum, is called a <b>finite mixture,</b> and in applications, an unqualified reference to a "mixture density" usually means a finite mixture. The case of a countably infinite set of components is covered formally by allowing <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=\infty \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\infty \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d9496db67ce03c5186bc1c5aebb3510940bf680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.257ex; width:6.687ex; height:1.676ex;" alt="{\displaystyle n=\infty \!}"></span> . </p> <div class="mw-heading mw-heading2"><h2 id="Uncountable_mixtures">Uncountable mixtures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=2" title="Edit section: Uncountable mixtures"><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/Compound_distribution" class="mw-redirect" title="Compound distribution">Compound distribution</a></div> <p>Where the set of component distributions is <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a>, the result is often called a <a href="/wiki/Compound_probability_distribution" title="Compound probability distribution">compound probability distribution</a>. The construction of such distributions has a formal similarity to that of mixture distributions, with either infinite summations or integrals replacing the finite summations used for finite mixtures. </p><p>Consider a probability density function <i>p</i>(<i>x</i>;<i>a</i>) for a variable <i>x</i>, parameterized by <i>a</i>. That is, for each value of <i>a</i> in some set <i>A</i>, <i>p</i>(<i>x</i>;<i>a</i>) is a probability density function with respect to <i>x</i>. Given a probability density function <i>w</i> (meaning that <i>w</i> is nonnegative and integrates to 1), the function </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)=\int _{A}\,w(a)\,p(x;a)\,da}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>w</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{A}\,w(a)\,p(x;a)\,da}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1ee43b4c626bbe16a9ac1289ca7dfe973d3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.543ex; height:5.676ex;" alt="{\displaystyle f(x)=\int _{A}\,w(a)\,p(x;a)\,da}"></span></dd></dl> <p>is again a probability density function for <i>x</i>. A similar integral can be written for the cumulative distribution function. Note that the formulae here reduce to the case of a finite or infinite mixture if the density <i>w</i> is allowed to be a <a href="/wiki/Generalized_function" title="Generalized function">generalized function</a> representing the "derivative" of the cumulative distribution function of a <a href="/wiki/Discrete_distribution" class="mw-redirect" title="Discrete distribution">discrete distribution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mixtures_within_a_parametric_family">Mixtures within a parametric family</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=3" title="Edit section: Mixtures within a parametric family"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mixture components are often not arbitrary probability distributions, but instead are members of a <a href="/wiki/Parametric_family" title="Parametric family">parametric family</a> (such as normal distributions), with different values for a parameter or parameters. In such cases, assuming that it exists, the density can be written in the form of a sum 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 f(x;a_{1},\ldots ,a_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;a_{1},\ldots ,a_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e293b567f2bab7a216a8ade0945d088c50bb493f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.812ex; height:6.843ex;" alt="{\displaystyle f(x;a_{1},\ldots ,a_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i})}"></span></dd></dl> <p>for one parameter, or </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;a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i},b_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i},b_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5216f7468590b2cce841ce78dc8cac70c4d2b29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.124ex; height:6.843ex;" alt="{\displaystyle f(x;a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i},b_{i})}"></span></dd></dl> <p>for two parameters, and so forth. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Convexity">Convexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=5" title="Edit section: Convexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of probability density functions is not necessarily a probability density, since it may be negative or it may integrate to something other than 1. However, a <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a> of probability density functions preserves both of these properties (non-negativity and integrating to 1), and thus mixture densities are themselves probability density functions. </p> <div class="mw-heading mw-heading3"><h3 id="Moments">Moments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=6" title="Edit section: Moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub> denote random variables from the <i>n</i> component distributions, and let <i>X</i> denote a random variable from the mixture distribution. Then, for any function <i>H</i>(·) for which <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} [H(X_{i})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [H(X_{i})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd42a0b12d4585fa6fd61a868ceef7e433517dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.473ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [H(X_{i})]}"></span> exists, and assuming that the component densities <i>p<sub>i</sub></i>(<i>x</i>) exist, </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} [H(X)]&=\int _{-\infty }^{\infty }H(x)\sum _{i=1}^{n}w_{i}p_{i}(x)\,dx\\&=\sum _{i=1}^{n}w_{i}\int _{-\infty }^{\infty }p_{i}(x)H(x)\,dx=\sum _{i=1}^{n}w_{i}\operatorname {E} [H(X_{i})].\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>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [H(X)]&=\int _{-\infty }^{\infty }H(x)\sum _{i=1}^{n}w_{i}p_{i}(x)\,dx\\&=\sum _{i=1}^{n}w_{i}\int _{-\infty }^{\infty }p_{i}(x)H(x)\,dx=\sum _{i=1}^{n}w_{i}\operatorname {E} [H(X_{i})].\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ebe639940c5f4b8d862fbf091fc2f24271fa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:57.061ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [H(X)]&=\int _{-\infty }^{\infty }H(x)\sum _{i=1}^{n}w_{i}p_{i}(x)\,dx\\&=\sum _{i=1}^{n}w_{i}\int _{-\infty }^{\infty }p_{i}(x)H(x)\,dx=\sum _{i=1}^{n}w_{i}\operatorname {E} [H(X_{i})].\end{aligned}}}"></span></dd></dl> <p>The <i>j</i>th moment about zero (i.e. choosing <span class="nowrap"><i>H</i>(<i>x</i>) = <i>x<sup>j</sup></i></span>) is simply a weighted average of the <i>j</i>th moments of the components. Moments about the mean <span class="nowrap"><i>H</i>(<i>x</i>) = (<i>x − μ</i>)<sup><i>j</i></sup></span> involve a binomial expansion:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></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 {\begin{aligned}\operatorname {E} [(X-\mu )^{j}]&=\sum _{i=1}^{n}w_{i}\operatorname {E} [(X_{i}-\mu _{i}+\mu _{i}-\mu )^{j}]\\&=\sum _{i=1}^{n}w_{i}\sum _{k=0}^{j}\left({\begin{array}{c}j\\k\end{array}}\right)(\mu _{i}-\mu )^{j-k}\operatorname {E} [(X_{i}-\mu _{i})^{k}],\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> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [(X-\mu )^{j}]&=\sum _{i=1}^{n}w_{i}\operatorname {E} [(X_{i}-\mu _{i}+\mu _{i}-\mu )^{j}]\\&=\sum _{i=1}^{n}w_{i}\sum _{k=0}^{j}\left({\begin{array}{c}j\\k\end{array}}\right)(\mu _{i}-\mu )^{j-k}\operatorname {E} [(X_{i}-\mu _{i})^{k}],\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc44ecc2d7e7d782fc6aaed8df1635784413436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.461ex; margin-bottom: -0.211ex; width:57.645ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [(X-\mu )^{j}]&=\sum _{i=1}^{n}w_{i}\operatorname {E} [(X_{i}-\mu _{i}+\mu _{i}-\mu )^{j}]\\&=\sum _{i=1}^{n}w_{i}\sum _{k=0}^{j}\left({\begin{array}{c}j\\k\end{array}}\right)(\mu _{i}-\mu )^{j-k}\operatorname {E} [(X_{i}-\mu _{i})^{k}],\end{aligned}}}"></span></dd></dl> <p>where <i>μ<sub>i</sub></i> denotes the mean of the <i>i</i>th component. </p><p>In the case of a mixture of one-dimensional distributions with weights <i>w<sub>i</sub></i>, means <i>μ<sub>i</sub></i> and variances <i>σ<sub>i</sub></i><sup>2</sup>, the total mean and variance will be: </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} [X]=\mu =\sum _{i=1}^{n}w_{i}\mu _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>μ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=\mu =\sum _{i=1}^{n}w_{i}\mu _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6515a2d9938b25ad5469936ebcf1aa8781b8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.509ex; height:6.843ex;" alt="{\displaystyle \operatorname {E} [X]=\mu =\sum _{i=1}^{n}w_{i}\mu _{i},}"></span></dd></dl> <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} [(X-\mu )^{2}]&=\sigma ^{2}\\&=\operatorname {E} [X^{2}]-\mu ^{2}&(\mathrm {standard} \ \mathrm {variance} \ \mathrm {reformulation} )\\&=\left(\sum _{i=1}^{n}w_{i}(\operatorname {E} [X_{i}^{2}])\right)-\mu ^{2}\\&=\sum _{i=1}^{n}w_{i}(\sigma _{i}^{2}+\mu _{i}^{2})-\mu ^{2}&(\mathrm {from} \ \sigma _{i}^{2}=\operatorname {E} [X_{i}^{2}]-\mu _{i}^{2},\mathrm {therefore} \,\operatorname {E} [X_{i}^{2}]=\sigma _{i}^{2}+\mu _{i}^{2}.)\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> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>−</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [(X-\mu )^{2}]&=\sigma ^{2}\\&=\operatorname {E} [X^{2}]-\mu ^{2}&(\mathrm {standard} \ \mathrm {variance} \ \mathrm {reformulation} )\\&=\left(\sum _{i=1}^{n}w_{i}(\operatorname {E} [X_{i}^{2}])\right)-\mu ^{2}\\&=\sum _{i=1}^{n}w_{i}(\sigma _{i}^{2}+\mu _{i}^{2})-\mu ^{2}&(\mathrm {from} \ \sigma _{i}^{2}=\operatorname {E} [X_{i}^{2}]-\mu _{i}^{2},\mathrm {therefore} \,\operatorname {E} [X_{i}^{2}]=\sigma _{i}^{2}+\mu _{i}^{2}.)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c042196b281fc86b4d20ac9b14e7f0c51a2ec81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:96.044ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [(X-\mu )^{2}]&=\sigma ^{2}\\&=\operatorname {E} [X^{2}]-\mu ^{2}&(\mathrm {standard} \ \mathrm {variance} \ \mathrm {reformulation} )\\&=\left(\sum _{i=1}^{n}w_{i}(\operatorname {E} [X_{i}^{2}])\right)-\mu ^{2}\\&=\sum _{i=1}^{n}w_{i}(\sigma _{i}^{2}+\mu _{i}^{2})-\mu ^{2}&(\mathrm {from} \ \sigma _{i}^{2}=\operatorname {E} [X_{i}^{2}]-\mu _{i}^{2},\mathrm {therefore} \,\operatorname {E} [X_{i}^{2}]=\sigma _{i}^{2}+\mu _{i}^{2}.)\end{aligned}}}"></span></dd></dl> <p>These relations highlight the potential of mixture distributions to display non-trivial higher-order moments such as <a href="/wiki/Skewness" title="Skewness">skewness</a> and <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a> (<a href="/wiki/Fat_tail" class="mw-redirect" title="Fat tail">fat tails</a>) and multi-modality, even in the absence of such features within the components themselves. Marron and Wand (1992) give an illustrative account of the flexibility of this framework.<sup id="cite_ref-Marron92_2-0" class="reference"><a href="#cite_note-Marron92-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Modes">Modes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=7" title="Edit section: Modes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The question of <a href="/wiki/Multimodal_distribution" title="Multimodal distribution">multimodality</a> is simple for some cases, such as mixtures of <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distributions</a>: all such mixtures are <a href="/wiki/Unimodality" title="Unimodality">unimodal</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, for the case of mixtures of <a href="/wiki/Normal_distribution" title="Normal distribution">normal distributions</a>, it is a complex one. Conditions for the number of modes in a multivariate normal mixture are explored by Ray & Lindsay<sup id="cite_ref-RayLindsay_4-0" class="reference"><a href="#cite_note-RayLindsay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> extending earlier work on univariate<sup id="cite_ref-Robertson1969_5-0" class="reference"><a href="#cite_note-Robertson1969-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Behboodian1970_6-0" class="reference"><a href="#cite_note-Behboodian1970-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and multivariate<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> distributions. </p><p>Here the problem of evaluation of the modes of an <i>n</i> component mixture in a <i>D</i> dimensional space is reduced to identification of critical points (local minima, maxima and <a href="/wiki/Saddle_point" title="Saddle point">saddle points</a>) on a <a href="/wiki/Manifold" title="Manifold">manifold</a> referred to as the <a href="/wiki/Ridge_(differential_geometry)" title="Ridge (differential geometry)">ridgeline surface</a>, which is the image of the ridgeline function </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 x^{*}(\alpha )=\left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\right]^{-1}\times \left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\mu _{i}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}(\alpha )=\left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\right]^{-1}\times \left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\mu _{i}\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8d8747d9b482096b6a40d509a7b81826d2f4a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.689ex; height:8.009ex;" alt="{\displaystyle x^{*}(\alpha )=\left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\right]^{-1}\times \left[\sum _{i=1}^{n}\alpha _{i}\Sigma _{i}^{-1}\mu _{i}\right],}"></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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> belongs to 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-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"></span>-dimensional standard <a href="/wiki/Simplex" title="Simplex">simplex</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 {S}}_{n}=\{\alpha \in \mathbb {R} ^{n}:\alpha _{i}\in [0,1],\sum _{i=1}^{n}\alpha _{i}=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</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>:</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{n}=\{\alpha \in \mathbb {R} ^{n}:\alpha _{i}\in [0,1],\sum _{i=1}^{n}\alpha _{i}=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3585c60b3e77e2d4fa2fbadf31481408ff076c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.317ex; height:6.843ex;" alt="{\displaystyle {\mathcal {S}}_{n}=\{\alpha \in \mathbb {R} ^{n}:\alpha _{i}\in [0,1],\sum _{i=1}^{n}\alpha _{i}=1\}}"></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 \Sigma _{i}\in R^{D\times D},\,\mu _{i}\in R^{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo>×</mo> <mi>D</mi> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{i}\in R^{D\times D},\,\mu _{i}\in R^{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8311af202340f85e5e8434fa2fc476e03282942b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.135ex; height:3.176ex;" alt="{\displaystyle \Sigma _{i}\in R^{D\times D},\,\mu _{i}\in R^{D}}"></span> correspond to the covariance and mean of the <i>i</i><sup>th</sup> component. Ray & Lindsay<sup id="cite_ref-RayLindsay_4-1" class="reference"><a href="#cite_note-RayLindsay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> consider the case in which <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-1<D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1<D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3b00b7b15f4cf01386e86b181e6cd9435d09c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.42ex; height:2.343ex;" alt="{\displaystyle n-1<D}"></span> showing a one-to-one correspondence of modes of the mixture and those on the <b>ridge elevation 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 h(\alpha )=q(x^{*}(\alpha ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(\alpha )=q(x^{*}(\alpha ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28cbeacb7555889d1b5d668b4656ec35b0cd529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.294ex; height:2.843ex;" alt="{\displaystyle h(\alpha )=q(x^{*}(\alpha ))}"></span> thus one may identify the modes by solving <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 {\frac {dh(\alpha )}{d\alpha }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dh(\alpha )}{d\alpha }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93da731a0eee29915f01c912874de8fb573d8a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.949ex; height:5.843ex;" alt="{\displaystyle {\frac {dh(\alpha )}{d\alpha }}=0}"></span> with respect to <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and determining the value <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^{*}(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c82e4fa5af46df3946c873005fc525ddf358cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.681ex; height:2.843ex;" alt="{\displaystyle x^{*}(\alpha )}"></span>. </p><p>Using graphical tools, the potential multi-modality of mixtures with number of components <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\in \{2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \{2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b60115668c66eab840dd7bafa95d0f3a3ea4a19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.919ex; height:2.843ex;" alt="{\displaystyle n\in \{2,3\}}"></span> is demonstrated; in particular it is shown that the number of modes may exceed <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> and that the modes may not be coincident with the component means. For two components they develop a graphical tool for analysis by instead solving the aforementioned differential with respect to the first mixing weight <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 w_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6728d2b30f42f88b52281be5ae0584fdc9df64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle w_{1}}"></span> (which also determines the second mixing weight through <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 w_{2}=1-w_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{2}=1-w_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24732c29f4c86c3d48a379531cb33c118debb56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.538ex; height:2.509ex;" alt="{\displaystyle w_{2}=1-w_{1}}"></span>) and expressing the solutions as 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 \Pi (\alpha ),\,\alpha \in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>α</mi> <mo>∈</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 \Pi (\alpha ),\,\alpha \in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f055596b0f6f5961bd829503ef601cb602f096f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.442ex; height:2.843ex;" alt="{\displaystyle \Pi (\alpha ),\,\alpha \in [0,1]}"></span> so that the number and location of modes for a given value 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 w_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6728d2b30f42f88b52281be5ae0584fdc9df64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle w_{1}}"></span> corresponds to the number of intersections of the graph on the line <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 \Pi (\alpha )=w_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (\alpha )=w_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9afd712e1fed96b27da526a69c991e2dd580114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.857ex; height:2.843ex;" alt="{\displaystyle \Pi (\alpha )=w_{1}}"></span>. This in turn can be related to the number of oscillations of the graph and therefore to solutions 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 {\frac {d\Pi (\alpha )}{d\alpha }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\Pi (\alpha )}{d\alpha }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb3713c3b38b5a3c3c1b6b776bef9f1469fae42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.353ex; height:5.843ex;" alt="{\displaystyle {\frac {d\Pi (\alpha )}{d\alpha }}=0}"></span> leading to an explicit solution for the case of a two component mixture 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 \Sigma _{1}=\Sigma _{2}=\Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{1}=\Sigma _{2}=\Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c0ba514884531e9ec2e86d40fc5d5fd0989383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.34ex; height:2.509ex;" alt="{\displaystyle \Sigma _{1}=\Sigma _{2}=\Sigma }"></span> (sometimes called a <a href="/wiki/Homoscedastic" class="mw-redirect" title="Homoscedastic">homoscedastic</a> mixture) given 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 1-\alpha (1-\alpha )d_{M}(\mu _{1},\mu _{2},\Sigma )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha (1-\alpha )d_{M}(\mu _{1},\mu _{2},\Sigma )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12db0f209293e48e396ba257e79c133468bd1548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.48ex; height:3.176ex;" alt="{\displaystyle 1-\alpha (1-\alpha )d_{M}(\mu _{1},\mu _{2},\Sigma )^{2}}"></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 d_{M}(\mu _{1},\mu _{2},\Sigma )={\sqrt {(\mu _{2}-\mu _{1})^{T}\Sigma ^{-1}(\mu _{2}-\mu _{1})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{M}(\mu _{1},\mu _{2},\Sigma )={\sqrt {(\mu _{2}-\mu _{1})^{T}\Sigma ^{-1}(\mu _{2}-\mu _{1})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb04542b0386105cb01ad7e042ff4b035f620bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:43.581ex; height:4.843ex;" alt="{\displaystyle d_{M}(\mu _{1},\mu _{2},\Sigma )={\sqrt {(\mu _{2}-\mu _{1})^{T}\Sigma ^{-1}(\mu _{2}-\mu _{1})}}}"></span> is the <a href="/wiki/Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a> between <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 \mu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6899621035d3359b9c1c064739b54c7004e220d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{1}}"></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 \mu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f841461ae8f2eafec3fe879f7c061a73c2f7170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{2}}"></span>. </p><p>Since the above is quadratic it follows that in this instance there are at most two modes irrespective of the dimension or the weights. </p><p>For normal mixtures with general <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>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}"></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 D>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23dcfe58133b398c11d4fdc158bc8287529efd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle D>1}"></span>, a lower bound for the maximum number of possible modes, and – conditionally on the assumption that the maximum number is finite – an upper bound are known. For those combinations 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 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> 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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> for which the maximum number is known, it matches the lower bound.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Two_normal_distributions">Two normal distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=9" title="Edit section: Two normal distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Simple examples can be given by a mixture of two normal distributions. (See <a href="/wiki/Multimodal_distribution#Mixture_of_two_normal_distributions" title="Multimodal distribution">Multimodal distribution#Mixture of two normal distributions</a> for more details.) </p><p>Given an equal (50/50) mixture of two normal distributions with the same standard deviation and different means (<a href="/wiki/Homoscedastic" class="mw-redirect" title="Homoscedastic">homoscedastic</a>), the overall distribution will exhibit low <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a> relative to a single normal distribution – the means of the subpopulations fall on the shoulders of the overall distribution. If sufficiently separated, namely by twice the (common) standard deviation, so <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|\mu _{1}-\mu _{2}\right|>2\sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>></mo> <mn>2</mn> <mi>σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mu _{1}-\mu _{2}\right|>2\sigma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d110ea0832d91be4608285700017490d701b490f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle \left|\mu _{1}-\mu _{2}\right|>2\sigma ,}"></span> these form a <a href="/wiki/Bimodal_distribution" class="mw-redirect" title="Bimodal distribution">bimodal distribution</a>, otherwise it simply has a wide peak.<sup id="cite_ref-Schilling2002_9-0" class="reference"><a href="#cite_note-Schilling2002-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The variation of the overall population will also be greater than the variation of the two subpopulations (due to spread from different means), and thus exhibits <a href="/wiki/Overdispersion" title="Overdispersion">overdispersion</a> relative to a normal distribution with fixed variation <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 \sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec50432d2d0f5329dacde3a76c502563f6bda6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle \sigma ,}"></span> though it will not be overdispersed relative to a normal distribution with variation equal to variation of the overall population. </p><p>Alternatively, given two subpopulations with the same mean and different standard deviations, the overall population will exhibit high kurtosis, with a sharper peak and heavier tails (and correspondingly shallower shoulders) than a single distribution. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Bimodal.png" class="mw-file-description" title="Univariate mixture distribution, showing bimodal distribution"><img alt="Univariate mixture distribution, showing bimodal distribution" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Bimodal.png/120px-Bimodal.png" decoding="async" width="120" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Bimodal.png/180px-Bimodal.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Bimodal.png/240px-Bimodal.png 2x" data-file-width="350" data-file-height="184" /></a></span></div> <div class="gallerytext">Univariate mixture distribution, showing bimodal distribution</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Bimodal-bivariate-small.png" class="mw-file-description" title="Multivariate mixture distribution, showing four modes"><img alt="Multivariate mixture distribution, showing four modes" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bimodal-bivariate-small.png/120px-Bimodal-bivariate-small.png" decoding="async" width="120" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bimodal-bivariate-small.png/180px-Bimodal-bivariate-small.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Bimodal-bivariate-small.png/240px-Bimodal-bivariate-small.png 2x" data-file-width="403" data-file-height="298" /></a></span></div> <div class="gallerytext">Multivariate mixture distribution, showing four modes</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="A_normal_and_a_Cauchy_distribution">A normal and a Cauchy distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=10" title="Edit section: A normal and a Cauchy distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following example is adapted from Hampel,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> who credits <a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a>. </p><p>Consider the mixture distribution defined by </p> <dl><dd><span class="texhtml"><i>F</i>(<i>x</i>)   =   (1 − 10<sup>−10</sup>) (<a href="/wiki/Normal_distribution" title="Normal distribution">standard normal</a>) + 10<sup>−10</sup> (<a href="/wiki/Cauchy_distribution" title="Cauchy distribution">standard Cauchy</a>)</span>.</dd></dl> <p>The mean of <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> observations from <span class="texhtml"><i>F</i>(<i>x</i>)</span> behaves "normally" except for exorbitantly large samples, although the mean of <span class="texhtml"><i>F</i>(<i>x</i>)</span> does not even exist. </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=Mixture_distribution&action=edit&section=11" title="Edit section: Applications"><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">Further information: <a href="/wiki/Mixture_model" title="Mixture model">Mixture model</a></div> <p>Mixture densities are complicated densities expressible in terms of simpler densities (the mixture components), and are used both because they provide a good model for certain data sets (where different subsets of the data exhibit different characteristics and can best be modeled separately), and because they can be more mathematically tractable, because the individual mixture components can be more easily studied than the overall mixture density. </p><p>Mixture densities can be used to model a <a href="/wiki/Statistical_population" title="Statistical population">statistical population</a> with <a href="/wiki/Subpopulation" class="mw-redirect" title="Subpopulation">subpopulations</a>, where the mixture components are the densities on the subpopulations, and the weights are the proportions of each subpopulation in the overall population. </p><p>Mixture densities can also be used to model <a href="/wiki/Experimental_error" class="mw-redirect" title="Experimental error">experimental error</a> or contamination – one assumes that most of the samples measure the desired phenomenon, with some samples from a different, erroneous distribution. </p><p>Parametric statistics that assume no error often fail on such mixture densities – for example, statistics that assume normality often fail disastrously in the presence of even a few <a href="/wiki/Outliers" class="mw-redirect" title="Outliers">outliers</a> – and instead one uses <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistics</a>. </p><p>In <a href="/wiki/Meta-analysis" title="Meta-analysis">meta-analysis</a> of separate studies, <a href="/wiki/Study_heterogeneity" title="Study heterogeneity">study heterogeneity</a> causes distribution of results to be a mixture distribution, and leads to <a href="/wiki/Overdispersion" title="Overdispersion">overdispersion</a> of results relative to predicted error. For example, in a <a href="/wiki/Statistical_survey" class="mw-redirect" title="Statistical survey">statistical survey</a>, the <a href="/wiki/Margin_of_error" title="Margin of error">margin of error</a> (determined by sample size) predicts the <a href="/wiki/Sampling_error" title="Sampling error">sampling error</a> and hence dispersion of results on repeated surveys. The presence of study heterogeneity (studies have different <a href="/wiki/Sampling_bias" title="Sampling bias">sampling bias</a>) increases the dispersion relative to the margin of error. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Compound_distribution" class="mw-redirect" title="Compound distribution">Compound distribution</a></li> <li><a href="/wiki/Contaminated_normal_distribution" class="mw-redirect" title="Contaminated normal distribution">Contaminated normal distribution</a></li> <li><a href="/wiki/Convex_combination" title="Convex combination">Convex combination</a></li> <li><a href="/wiki/Giry_monad" title="Giry monad">Giry monad</a></li> <li><a href="/wiki/Expectation-maximization_algorithm" class="mw-redirect" title="Expectation-maximization algorithm">Expectation-maximization (EM) algorithm</a></li> <li>Not to be confused with: <a href="/wiki/List_of_convolutions_of_probability_distributions" title="List of convolutions of probability distributions">list of convolutions of probability distributions</a></li> <li><a href="/wiki/Product_distribution" class="mw-redirect" title="Product distribution">Product distribution</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mixture">Mixture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=13" title="Edit section: Mixture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mixture_(probability)" title="Mixture (probability)">Mixture (probability)</a></li> <li><a href="/wiki/Mixture_model" title="Mixture model">Mixture model</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Hierarchical_models">Hierarchical models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=14" title="Edit section: Hierarchical models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Hierarchical_Bayes_model" class="mw-redirect" title="Hierarchical Bayes model">Hierarchical Bayes model</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mixture_distribution&action=edit&section=15" title="Edit section: Notes"><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-1"><span class="mw-cite-backlink"><b><a 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"Is human height bimodal?". <i><a href="/wiki/The_American_Statistician" title="The American Statistician">The American Statistician</a></i>. <b>56</b> (3): 223–229. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1198%2F00031300265">10.1198/00031300265</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Is+human+height+bimodal%3F&rft.volume=56&rft.issue=3&rft.pages=223-229&rft.date=2002&rft_id=info%3Adoi%2F10.1198%2F00031300265&rft.aulast=Schilling&rft.aufirst=Mark+F.&rft.au=Watkins%2C+Ann+E.&rft.au=Watkins%2C+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHampel1998" class="citation cs2">Hampel, Frank (1998), "Is statistics too difficult?", <i>Canadian Journal of Statistics</i>, <b>26</b>: 497–513, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3315772">10.2307/3315772</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/20.500.11850%2F145503">20.500.11850/145503</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Statistics&rft.atitle=Is+statistics+too+difficult%3F&rft.volume=26&rft.pages=497-513&rft.date=1998&rft_id=info%3Ahdl%2F20.500.11850%2F145503&rft_id=info%3Adoi%2F10.2307%2F3315772&rft.aulast=Hampel&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span></span> </li> </ol></div></div> <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=Mixture_distribution&action=edit&section=16" title="Edit section: References"><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="CITEREFFrühwirth-Schnatter2006" class="citation cs2">Frühwirth-Schnatter, Sylvia (2006), <i>Finite Mixture and Markov Switching Models</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-2194-9" title="Special:BookSources/978-1-4419-2194-9"><bdi>978-1-4419-2194-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Mixture+and+Markov+Switching+Models&rft.pub=Springer&rft.date=2006&rft.isbn=978-1-4419-2194-9&rft.aulast=Fr%C3%BChwirth-Schnatter&rft.aufirst=Sylvia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLindsay1995" class="citation cs2">Lindsay, Bruce G. (1995), <i>Mixture models: theory, geometry and applications</i>, NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 5, Hayward, CA, USA: Institute of Mathematical Statistics, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-940600-32-3" title="Special:BookSources/0-940600-32-3"><bdi>0-940600-32-3</bdi></a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4153184">4153184</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mixture+models%3A+theory%2C+geometry+and+applications&rft.place=Hayward%2C+CA%2C+USA&rft.series=NSF-CBMS+Regional+Conference+Series+in+Probability+and+Statistics&rft.pub=Institute+of+Mathematical+Statistics&rft.date=1995&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4153184%23id-name%3DJSTOR&rft.isbn=0-940600-32-3&rft.aulast=Lindsay&rft.aufirst=Bruce+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeidel2010" class="citation cs2">Seidel, Wilfried (2010), "Mixture models", in Lovric, M. (ed.), <i>International Encyclopedia of Statistical Science</i>, Heidelberg: Springer, pp. 827–829, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0909.0389">0909.0389</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-04898-2">10.1007/978-3-642-04898-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-04898-2" title="Special:BookSources/978-3-642-04898-2"><bdi>978-3-642-04898-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mixture+models&rft.btitle=International+Encyclopedia+of+Statistical+Science&rft.place=Heidelberg&rft.pages=827-829&rft.pub=Springer&rft.date=2010&rft_id=info%3Aarxiv%2F0909.0389&rft_id=info%3Adoi%2F10.1007%2F978-3-642-04898-2&rft.isbn=978-3-642-04898-2&rft.aulast=Seidel&rft.aufirst=Wilfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYaoXiang2024" class="citation cs2">Yao, Weixin; Xiang, Sijia (2024), <a rel="nofollow" class="external text" href="https://www.routledge.com/Mixture-Models-Parametric-Semiparametric-and-New-Directions/Yao-Xiang/p/book/9780367481827"><i>Mixture Models: Parametric, Semiparametric, and New Directions</i></a>, Boca Raton, FL: Chapman & Hall/CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0367481827" title="Special:BookSources/978-0367481827"><bdi>978-0367481827</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mixture+Models%3A+Parametric%2C+Semiparametric%2C+and+New+Directions&rft.place=Boca+Raton%2C+FL&rft.pub=Chapman+%26+Hall%2FCRC+Press&rft.date=2024&rft.isbn=978-0367481827&rft.aulast=Yao&rft.aufirst=Weixin&rft.au=Xiang%2C+Sijia&rft_id=https%3A%2F%2Fwww.routledge.com%2FMixture-Models-Parametric-Semiparametric-and-New-Directions%2FYao-Xiang%2Fp%2Fbook%2F9780367481827&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMixture+distribution" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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