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<title>Moment-generating function</title>

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Moment-generating function

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In <a href="Probability_theory" title="Probability theory">probability theory</a> and <a href="Statistics" title="Statistics">statistics</a>, the moment-generating function of a real-valued <a href="Random_variable" title="Random variable">random variable</a> is an alternative specification of its <a href="Probability_distribution" title="Probability distribution">probability distribution</a>. Thus, it provides the basis of an alternative route to analytical results compared with working directly with <a href="Probability_density_function" title="Probability density function">probability density functions</a> or <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a>. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.

As its name implies, the moment-<a href="Generating_function" title="Generating function">generating function</a> can be used to compute a distribution’s <a href="Moment_(mathematics)" title="Moment (mathematics)">moments</a>: the n-th moment about 0 is the n-th derivative of the moment-generating function, evaluated at 0.

In addition to univariate real-valued distributions, moment-generating functions can also be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.

The moment-generating function of a real-valued distribution does not always exist, unlike the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a>. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a <a href="Random_variable" title="Random variable">random variable</a> with <a href="Cumulative_distribution_function" title="Cumulative distribution function">CDF</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{X}}</annotation>

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</math><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy">. The moment generating function (mgf) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{X}}</annotation>

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</math><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy">), denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\operatorname {E} \left[e^{tX}\right]}</annotation>

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provided this <a href="Expected_value" title="Expected value">expectation</a> exists for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> in some open <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of 0. That is, there is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h>0}">

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</math><img src="./cbddb7a5cca6170575e4e73e769fbb434c2a3d71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -h<t<h}">

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<annotation encoding="application/x-tex">{\displaystyle -h<t<h}</annotation>

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</math><img src="./482bb239e635eacef7df47ae2efa29356a4885f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.523ex; height:2.343ex;" alt="{\displaystyle -h<t<h}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{tX}\right]}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{tX}\right]}</annotation>

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</math><img src="./ad12410185c69f6ffc6889791e227412a323406d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.832ex; height:3.343ex;" alt="{\displaystyle \operatorname {E} \left[e^{tX}\right]}" loading="lazy"> exists. If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist.<a href="#cite_note-1">[1]</a>

In other words, the moment-generating function of X is the <a href="Expected_value" title="Expected value">expectation</a> of the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{tX}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{tX}}</annotation>

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</math><img src="./b20e8a2103c26e6e2602c9ec39c1acdb7a639ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.31ex; height:2.676ex;" alt="{\displaystyle e^{tX}}" loading="lazy">. More generally, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }}</annotation>

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</math><img src="./953715a22566a4b8eb109fc67b40c3cd37286d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.646ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }}" loading="lazy">, an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional <a href="Random_vector" class="mw-redirect" title="Random vector">random vector</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} }</annotation>

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</math><img src="./16ff63cc74a931900e79b3caacbae3fa8cc66845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} }" loading="lazy"> is a fixed vector, one uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} \cdot \mathbf {X} =\mathbf {t} ^{\mathrm {T} }\mathbf {X} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} \cdot \mathbf {X} =\mathbf {t} ^{\mathrm {T} }\mathbf {X} }</annotation>

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</math><img src="./d11dec264d6a5abada2090cc49eec5130b7326a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.314ex; height:2.676ex;" alt="{\displaystyle \mathbf {t} \cdot \mathbf {X} =\mathbf {t} ^{\mathrm {T} }\mathbf {X} }" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tX}">

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<annotation encoding="application/x-tex">{\displaystyle tX}</annotation>

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</math><img src="./a66280da7b129daf2209d27a60eb7bf5af56254a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.82ex; height:2.176ex;" alt="{\displaystyle tX}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle M_{\mathbf {X} }(\mathbf {t} ):=\operatorname {E} \left[e^{\mathbf {t} ^{\mathrm {T} }\mathbf {X} }\right].}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(0)}</annotation>

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</math><img src="./67bb96e8035b3098f87310b139377155b9e5dd08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.858ex; height:2.843ex;" alt="{\displaystyle M_{X}(0)}" loading="lazy"> always exists and is equal to 1. However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> or Fourier transform always exists (because it is the integral of a bounded function on a space of finite <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>), and for some purposes may be used instead.

The moment-generating function is so named because it can be used to find the moments of the distribution.<a href="#cite_note-2">[2]</a> The series expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{tX}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{tX}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle e^{tX}=1+tX+{\frac {t^{2}X^{2}}{2!}}+{\frac {t^{3}X^{3}}{3!}}+\cdots +{\frac {t^{n}X^{n}}{n!}}+\cdots .}</annotation>

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Hence,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{X}(t)&=\operatorname {E} [e^{tX}]\\[1ex]&=1+t\operatorname {E} [X]+{\frac {t^{2}\operatorname {E} [X^{2}]}{2!}}+{\frac {t^{3}\operatorname {E} [X^{3}]}{3!}}+\cdots +{\frac {t^{n}\operatorname {E} [X^{n}]}{n!}}+\cdots \\[1ex]&=1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+{\frac {t^{3}m_{3}}{3!}}+\cdots +{\frac {t^{n}m_{n}}{n!}}+\cdots ,\end{aligned}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{n}}</annotation>

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</math><img src="./ae8181c5cf15da902bbaa5c2291aeb8167fe78ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.259ex; height:2.009ex;" alt="{\displaystyle m_{n}}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th <a href="Moment_(mathematics)" title="Moment (mathematics)">moment</a>. Differentiating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

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</math><img src="./34fcd0a5942fd5fd42e0f8a7d15c301f9851fbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.535ex; height:2.843ex;" alt="{\displaystyle M_{X}(t)}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> times with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">, we obtain the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-th moment about the origin, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation>

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</math><img src="./95ec8e804f69706d3f5ad235f4f983220c8df7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}" loading="lazy">; see <a href="#Calculations_of_moments">§ Calculations of moments</a> below.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a continuous random variable, the following relation between its moment-generating function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

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</math><img src="./34fcd0a5942fd5fd42e0f8a7d15c301f9851fbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.535ex; height:2.843ex;" alt="{\displaystyle M_{X}(t)}" loading="lazy"> and the <a href="Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a> of its probability density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)={\mathcal {L}}\{f_{X}\}(-t),}</annotation>

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since the PDF's two-sided Laplace transform is given as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\{f_{X}\}(s)=\int _{-\infty }^{\infty }e^{-sx}f_{X}(x)\,dx,}</annotation>

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and the moment-generating function's definition expands (by the <a href="Law_of_the_unconscious_statistician" title="Law of the unconscious statistician">law of the unconscious statistician</a>) to

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\operatorname {E} \left[e^{tX}\right]=\int _{-\infty }^{\infty }e^{tx}f_{X}(x)\,dx.}</annotation>

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This is consistent with the characteristic function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> being a <a href="Wick_rotation" title="Wick rotation">Wick rotation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

</semantics>

</math><img src="./34fcd0a5942fd5fd42e0f8a7d15c301f9851fbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.535ex; height:2.843ex;" alt="{\displaystyle M_{X}(t)}" loading="lazy"> when the moment generating function exists, as the characteristic function of a continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of its probability density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)}</annotation>

</semantics>

</math><img src="./29310a010e7f7dfa33ba69bcf1ef9ec166d461dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)}" loading="lazy">, and in general when a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is of exponential order, the Fourier transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a Wick rotation of its two-sided Laplace transform in the region of convergence. See <a href="Fourier_transform#Laplace_transform" title="Fourier transform">the relation of the Fourier and Laplace transforms</a> for further information.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

Here are some examples of the moment-generating function and the characteristic function for comparison. It can be seen that the characteristic function is a <a href="Wick_rotation" title="Wick rotation">Wick rotation</a> of the moment-generating function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

</semantics>

<th>Distribution

</th>

<th>Moment-generating function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

</semantics>

</th>

<th>Characteristic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

</semantics>

</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy">

</th></tr>

<tr>

<td><a href="Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{a}}</annotation>

</semantics>

</math><img src="./5b0186141ec6e566dcfdb70100d7d1560c544d55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.676ex;" alt="{\displaystyle \delta _{a}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{ta}}</annotation>

</semantics>

</math><img src="./cdf2e2b36e95ce8ce6258c2161fcb74d40dae712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle e^{ta}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{ita}}</annotation>

</semantics>

</math><img src="./0e9dff8225c5131893eb10901e0a2ba287aeee51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.346ex; height:2.676ex;" alt="{\displaystyle e^{ita}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=1)=p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(X=1)=p}</annotation>

</semantics>

</math><img src="./82fec546358e0ee8a15b2e0651936d71450ba682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.064ex; height:2.843ex;" alt="{\displaystyle P(X=1)=p}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-p+pe^{t}}</annotation>

</semantics>

</math><img src="./94891a1d28b0a0050aacfec9e6f9d93511457a4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.092ex; height:2.843ex;" alt="{\displaystyle 1-p+pe^{t}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-p+pe^{it}}</annotation>

</semantics>

</math><img src="./beb23d2b2724ba1f0d9239e694c2fbc879e8feb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.659ex; height:3.009ex;" alt="{\displaystyle 1-p+pe^{it}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Binomial_distribution" title="Binomial distribution">Binomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n,p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(n,p)}</annotation>

</semantics>

</math><img src="./58280f6b0f1a1b474a7047c07943f908e775aa71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle B(n,p)}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-p+pe^{t}\right)^{n}}</annotation>

</semantics>

</math><img src="./af3bdd76bd88de6ccf4488b4c264f2f279ddb4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.44ex; height:3.343ex;" alt="{\displaystyle \left(1-p+pe^{t}\right)^{n}}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-p+pe^{it}\right)^{n}}</annotation>

</semantics>

</math><img src="./dc2a7e732602636b136d66bd3dbc03f66c09caef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.007ex; height:3.343ex;" alt="{\displaystyle \left(1-p+pe^{it}\right)^{n}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Geometric_distribution" title="Geometric distribution">Geometric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-p)^{k}\,p}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-p)^{k}\,p}</annotation>

</semantics>

</math><img src="./d754a23180123ed647df290ed8c1e429bcf595e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.627ex; height:3.176ex;" alt="{\displaystyle (1-p)^{k}\,p}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {p}{1-(1-p)e^{t}}},~t<-\ln(1-p)}</annotation>

</semantics>

</math><img src="./d0d3bf30e7763882976f45ebf223500c2014c6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.399ex; height:5.676ex;" alt="{\displaystyle {\frac {p}{1-(1-p)e^{t}}},~t<-\ln(1-p)}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {p}{1-(1-p)\,e^{it}}}}</annotation>

</semantics>

</math><img src="./3e4831a35c938e19709b5040ecfba320ec7d460e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.685ex; height:5.676ex;" alt="{\displaystyle {\frac {p}{1-(1-p)\,e^{it}}}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {NB} (r,p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {NB} (r,p)}</annotation>

</semantics>

</math><img src="./63daf5a88b1753d0f258bf3476f33a443dbebf42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.45ex; height:2.843ex;" alt="{\displaystyle \operatorname {NB} (r,p)}" loading="lazy">

</td>

<msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{1-e^{t}+pe^{t}}}\right)^{r},~t<-\ln(1-p)}</annotation>

</semantics>

</math><img src="./7c12a6d00255379799ed94b2169ca7f9febca140.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.731ex; height:6.176ex;" alt="{\displaystyle \left({\frac {p}{1-e^{t}+pe^{t}}}\right)^{r},~t<-\ln(1-p)}" loading="lazy">

</td>

<msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{1-e^{it}+pe^{it}}}\right)^{r}}</annotation>

</semantics>

</math><img src="./b81fa242550f41da047f5d1b4c3006807fd9b7ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.197ex; height:6.176ex;" alt="{\displaystyle \left({\frac {p}{1-e^{it}+pe^{it}}}\right)^{r}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Poisson_distribution" title="Poisson distribution">Poisson</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Pois} (\lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Pois} (\lambda )}</annotation>

</semantics>

</math><img src="./2171745e86cc9d5c28f0e2595db6b5532fe8bb47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.473ex; height:2.843ex;" alt="{\displaystyle \operatorname {Pois} (\lambda )}" loading="lazy">

</td>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\lambda (e^{t}-1)}}</annotation>

</semantics>

</math><img src="./bbe62d654f25b81cd581f87eacb7590938980c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.066ex; height:3.009ex;" alt="{\displaystyle e^{\lambda (e^{t}-1)}}" loading="lazy">

</td>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\lambda (e^{it}-1)}}</annotation>

</semantics>

</math><img src="./a0916d11e627713428da6413c6e3d705032d8c16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.527ex; height:3.009ex;" alt="{\displaystyle e^{\lambda (e^{it}-1)}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">Uniform (continuous)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {U} (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {U} (a,b)}</annotation>

</semantics>

</math><img src="./db682b0f09abad1921446433c0a4f82ae6ea3e90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.814ex; height:2.843ex;" alt="{\displaystyle \operatorname {U} (a,b)}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{tb}-e^{ta}}{t(b-a)}}}</annotation>

</semantics>

</math><img src="./04184b82f68e8db51176f0c2a93d7d365cdd7774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.07ex; height:6.509ex;" alt="{\displaystyle {\frac {e^{tb}-e^{ta}}{t(b-a)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{itb}-e^{ita}}{it(b-a)}}}</annotation>

</semantics>

</math><img src="./e2accc12a5cda98adf7a55f8273655655a6b1290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.205ex; height:6.509ex;" alt="{\displaystyle {\frac {e^{itb}-e^{ita}}{it(b-a)}}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Discrete_uniform_distribution" title="Discrete uniform distribution">Uniform (discrete)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DU} (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {DU} (a,b)}</annotation>

</semantics>

</math><img src="./f0f64e64ec6221ed153b26c15ebfc56e9e0fe3c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.589ex; height:2.843ex;" alt="{\displaystyle \operatorname {DU} (a,b)}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{at}-e^{(b+1)t}}{(b-a+1)(1-e^{t})}}}</annotation>

</semantics>

</math><img src="./aeb0e20f7ca280d2ae33d4c4282f9e678a5b253d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.438ex; height:6.509ex;" alt="{\displaystyle {\frac {e^{at}-e^{(b+1)t}}{(b-a+1)(1-e^{t})}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{ait}-e^{(b+1)it}}{(b-a+1)(1-e^{it})}}}</annotation>

</semantics>

</math><img src="./3ff28f45bf2cccc684af52529b14b7f42b6c2e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.005ex; height:6.509ex;" alt="{\displaystyle {\frac {e^{ait}-e^{(b+1)it}}{(b-a+1)(1-e^{it})}}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Laplace_distribution" title="Laplace distribution">Laplace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mu ,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\mu ,b)}</annotation>

</semantics>

</math><img src="./d0db0a9cd3e0514315bdabe70fffbe3c8e40b1d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.825ex; height:2.843ex;" alt="{\displaystyle L(\mu ,b)}" loading="lazy">

</td>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{t\mu }}{1-b^{2}t^{2}}},~|t|<1/b}</annotation>

</semantics>

</math><img src="./496ee0bda5076092b11c1bbac11c3380606cfb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.953ex; height:6.009ex;" alt="{\displaystyle {\frac {e^{t\mu }}{1-b^{2}t^{2}}},~|t|<1/b}" loading="lazy">

</td>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e^{it\mu }}{1+b^{2}t^{2}}}}</annotation>

</semantics>

</math><img src="./8581ab3135de1bbf38a9c50030cb180d5fee8c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.785ex; height:6.176ex;" alt="{\displaystyle {\frac {e^{it\mu }}{1+b^{2}t^{2}}}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Normal_distribution" title="Normal distribution">Normal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mu ,\sigma ^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(\mu ,\sigma ^{2})}</annotation>

</semantics>

</math><img src="./3cbd76720b12f0428a8bf1460b7a67cf2f5f3817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.693ex; height:3.176ex;" alt="{\displaystyle N(\mu ,\sigma ^{2})}" loading="lazy">

</td>

<msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{t\mu +\sigma ^{2}t^{2}/2}}</annotation>

</semantics>

</math><img src="./29960263c869b7f531fa70b66a1f4f0fb3bf5e4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.021ex; height:3.009ex;" alt="{\displaystyle e^{t\mu +\sigma ^{2}t^{2}/2}}" loading="lazy">

</td>

<msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{it\mu -\sigma ^{2}t^{2}/2}}</annotation>

</semantics>

</math><img src="./a70421573b25ebd081206dcf9a52569d95bbe7e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.588ex; height:3.009ex;" alt="{\displaystyle e^{it\mu -\sigma ^{2}t^{2}/2}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{k}^{2}}</annotation>

</semantics>

</math><img src="./cf08a208590f50934d624d2dc1e528a393faa4d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.544ex; height:3.176ex;" alt="{\displaystyle \chi _{k}^{2}}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\left(1-2t\right)}^{-k/2},~t<1/2}</annotation>

</semantics>

</math><img src="./7993169372ae6b980d7d4c6428c98b93f70d7c8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.865ex; height:3.509ex;" alt="{\displaystyle {\left(1-2t\right)}^{-k/2},~t<1/2}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\left(1-2it\right)}^{-{k}/{2}}}</annotation>

</semantics>

</math><img src="./f21e871b997ac5cd7b24c577fcc353ca517ef4d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.628ex; height:3.509ex;" alt="{\displaystyle {\left(1-2it\right)}^{-{k}/{2}}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral chi-squared</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}^{2}(\lambda )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{k}^{2}(\lambda )}</annotation>

</semantics>

</math><img src="./a9e18e730f74e5c783c65a2f2a7b989b7fc135f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.708ex; height:3.176ex;" alt="{\displaystyle \chi _{k}^{2}(\lambda )}" loading="lazy">

</td>

<msup>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\lambda t/(1-2t)}{\left(1-2t\right)}^{-k/2}}</annotation>

</semantics>

</math><img src="./8dcc0768387cf4a7d6b5b9b2dafe693ad16193b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.311ex; height:3.509ex;" alt="{\displaystyle e^{\lambda t/(1-2t)}{\left(1-2t\right)}^{-k/2}}" loading="lazy">

</td>

<msup>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\lambda t/(1-2it)}{\left(1-2it\right)}^{-k/2}}</annotation>

</semantics>

</math><img src="./0bf11f20d2e8d38d6556ad19982bbc618189cdf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.248ex; height:3.509ex;" alt="{\displaystyle e^{i\lambda t/(1-2it)}{\left(1-2it\right)}^{-k/2}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Gamma_distribution" title="Gamma distribution">Gamma</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (k,{\tfrac {1}{\theta }})}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (k,{\tfrac {1}{\theta }})}</annotation>

</semantics>

</math><img src="./48169a22ed7c8d68b85e9766959d0a3773ce2960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.165ex; height:3.676ex;" alt="{\displaystyle \Gamma (k,{\tfrac {1}{\theta }})}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\left(1-t\theta \right)}^{-k},~t<{\tfrac {1}{\theta }}}</annotation>

</semantics>

</math><img src="./270e5d0762c7fe78ea0e99a43d19c54e8b4f4260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.32ex; height:3.843ex;" alt="{\displaystyle {\left(1-t\theta \right)}^{-k},~t<{\tfrac {1}{\theta }}}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\left(1-it\theta \right)}^{-k}}</annotation>

</semantics>

</math><img src="./261d282ce6ed9d1731631e40397a9ca89e2132bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.912ex; height:3.343ex;" alt="{\displaystyle {\left(1-it\theta \right)}^{-k}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Exponential_distribution" title="Exponential distribution">Exponential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Exp} (\lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Exp} (\lambda )}</annotation>

</semantics>

</math><img src="./3c85a4db2e96aa5263e7a9ca4d6f33f17b56fe6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle \operatorname {Exp} (\lambda )}" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-t\lambda ^{-1}\right)^{-1},~t<\lambda }</annotation>

</semantics>

</math><img src="./cf65f552e12e2179dfabf582ce3877e61a8c1f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.901ex; height:3.843ex;" alt="{\displaystyle \left(1-t\lambda ^{-1}\right)^{-1},~t<\lambda }" loading="lazy">

</td>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-it\lambda ^{-1}\right)^{-1}}</annotation>

</semantics>

</math><img src="./ba6d777c65ce24f0797bdac2084f805cfc515589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.796ex; height:3.843ex;" alt="{\displaystyle \left(1-it\lambda ^{-1}\right)^{-1}}" loading="lazy">

</td></tr>

<tr>

</td>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}</annotation>

</semantics>

</math><img src="./b97b0e33f3134c2fc5c484016ab8e03e18d85481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.32ex; height:7.509ex;" alt="{\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}" loading="lazy">

</td>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!}</annotation>

</semantics>

</math><img src="./d05780b82c8372e644a400af4e75ecd83d454f53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:16.436ex; height:2.843ex;" alt="{\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!}" loading="lazy"> (see <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">Confluent hypergeometric function</a>)

</td></tr>

<tr>

<td><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mathbf {\mu } ,\mathbf {\Sigma } )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(\mathbf {\mu } ,\mathbf {\Sigma } )}</annotation>

</semantics>

</math><img src="./496ec3861a1eb0a24914f5e838a2cc2ae0f1bd6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.24ex; height:2.843ex;" alt="{\displaystyle N(\mathbf {\mu } ,\mathbf {\Sigma } )}" loading="lazy">

</td>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left[\mathbf {t} ^{\mathrm {T} }\left({\boldsymbol {\mu }}+{\tfrac {1}{2}}{\boldsymbol {\Sigma }}\mathbf {t} \right)\right]}</annotation>

</semantics>

</math><img src="./12c98412167dac079ac9f2d0ec5c362148b1a2a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.582ex; height:3.509ex;" alt="{\displaystyle \exp \left[\mathbf {t} ^{\mathrm {T} }\left({\boldsymbol {\mu }}+{\tfrac {1}{2}}{\boldsymbol {\Sigma }}\mathbf {t} \right)\right]}" loading="lazy">

</td>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left[\mathbf {t} ^{\mathrm {T} }\left(i{\boldsymbol {\mu }}-{\tfrac {1}{2}}{\boldsymbol {\Sigma }}\mathbf {t} \right)\right]}</annotation>

</semantics>

</math><img src="./1e70a6b9df258b2dfb1fc7bbfdd5ae81d8df9774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.384ex; height:3.509ex;" alt="{\displaystyle \exp \left[\mathbf {t} ^{\mathrm {T} }\left(i{\boldsymbol {\mu }}-{\tfrac {1}{2}}{\boldsymbol {\Sigma }}\mathbf {t} \right)\right]}" loading="lazy">

</td></tr>

<tr>

<td><a href="Cauchy_distribution" title="Cauchy distribution">Cauchy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cauchy} (\mu ,\theta )}">

<mi>Cauchy</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cauchy} (\mu ,\theta )}</annotation>

</semantics>

</math><img src="./f2b53b1b618c8bdacf28231195d15d98937acf6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.021ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cauchy} (\mu ,\theta )}" loading="lazy">

</td>

<td><a href="Indeterminate_form" title="Indeterminate form">Does not exist</a>

</td>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{it\mu -\theta |t|}}</annotation>

</semantics>

</math><img src="./a786c1c23531b1e55cc6c3ca315b4f4769181cb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:2.843ex;" alt="{\displaystyle e^{it\mu -\theta |t|}}" loading="lazy">

</td></tr>

<tr>

<td><a href="Multivariate_Cauchy_distribution" class="mw-redirect" title="Multivariate Cauchy distribution">Multivariate Cauchy</a>

<mi>MultiCauchy</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {MultiCauchy} (\mu ,\Sigma )}</annotation>

</semantics>

</math><img src="./c21d23adfb2163b1463577b8468283ea07e1d2f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.23ex; height:2.843ex;" alt="{\displaystyle \operatorname {MultiCauchy} (\mu ,\Sigma )}" loading="lazy"><a href="#cite_note-3">[3]</a>

</td>

<td>Does not exist

</td>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msqrt>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(i\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\mu }}-{\sqrt {\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\mathbf {t} }}\right)}</annotation>

</semantics>

</math><img src="./bd3ead01663e90651def071b66e793fbd00fb183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.827ex; height:4.843ex;" alt="{\displaystyle \exp \left(i\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\mu }}-{\sqrt {\mathbf {t} ^{\mathrm {T} }{\boldsymbol {\Sigma }}\mathbf {t} }}\right)}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2></div>

The moment-generating function is the expectation of a function of the random variable, it can be written as:

<ul><li>For a discrete <a href="Probability_mass_function" title="Probability mass function">probability mass function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=\sum _{i=0}^{\infty }e^{tx_{i}}\,p_{i}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\sum _{i=0}^{\infty }e^{tx_{i}}\,p_{i}}</annotation>

</semantics>

</math><img src="./ad439f418db2a4897fb6c23e7747fe7feeceb5e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.207ex; height:6.843ex;" alt="{\displaystyle M_{X}(t)=\sum _{i=0}^{\infty }e^{tx_{i}}\,p_{i}}" loading="lazy"></li>

<li>For a continuous <a href="Probability_density_function" title="Probability density function">probability density function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx}</annotation>

</semantics>

</math><img src="./67bd9e22c3642863a8e22367dd7b461e0c97a268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.667ex; height:6.009ex;" alt="{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx}" loading="lazy"></li>

<li>In the general case: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}\,dF(x)}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}\,dF(x)}</annotation>

</semantics>

</math><img src="./a1c01f220becd870d525aad5dd549b0ed9a33796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.8ex; height:6.009ex;" alt="{\displaystyle M_{X}(t)=\int _{-\infty }^{\infty }e^{tx}\,dF(x)}" loading="lazy">, using the <a href="Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>. This is simply the <a href="Laplace-Stieltjes_transform" class="mw-redirect" title="Laplace-Stieltjes transform">Laplace-Stieltjes transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

Note that for the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has a continuous <a href="Probability_density_function" title="Probability density function">probability density function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(-t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(-t)}</annotation>

</semantics>

</math><img src="./e5374386463a31864730a75f330e8148bcc41485.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.343ex; height:2.843ex;" alt="{\displaystyle M_{X}(-t)}" loading="lazy"> is the <a href="Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

<mtr>

<mtd>

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<mtd>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{X}(t)&=\int _{-\infty }^{\infty }e^{tx}f(x)\,dx\\[1ex]&=\int _{-\infty }^{\infty }\left(1+tx+{\frac {t^{2}x^{2}}{2!}}+\cdots +{\frac {t^{n}x^{n}}{n!}}+\cdots \right)f(x)\,dx\\[1ex]&=1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+\cdots +{\frac {t^{n}m_{n}}{n!}}+\cdots ,\end{aligned}}}</annotation>

</semantics>

</math>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{n}}</annotation>

</semantics>

</math><img src="./ae8181c5cf15da902bbaa5c2291aeb8167fe78ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.259ex; height:2.009ex;" alt="{\displaystyle m_{n}}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Linear_transformations_of_random_variables">Linear transformations of random variables</h3></div>

If random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has moment generating function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

</semantics>

</math><img src="./34fcd0a5942fd5fd42e0f8a7d15c301f9851fbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.535ex; height:2.843ex;" alt="{\displaystyle M_{X}(t)}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha X+\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha X+\beta }</annotation>

</semantics>

</math><img src="./1795c6cfe1b52140334d18a0834bccfaf06359d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.64ex; height:2.509ex;" alt="{\displaystyle \alpha X+\beta }" loading="lazy"> has moment generating function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)}">

<msub>

</mrow>

</msub>

<msup>

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</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)}</annotation>

</semantics>

</math><img src="./b563c312967bcf3e33bdc3fc4558c75a7174d4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.78ex; height:3.343ex;" alt="{\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)}" loading="lazy">

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

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</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha X+\beta }(t)=\operatorname {E} \left[e^{(\alpha X+\beta )t}\right]=e^{\beta t}\operatorname {E} \left[e^{\alpha Xt}\right]=e^{\beta t}M_{X}(\alpha t)}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Linear_combination_of_independent_random_variables">Linear combination of independent random variables</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S_{n}=\sum _{i=1}^{n}a_{i}X_{i}}">

<msub>

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</munderover>

<msub>

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</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle S_{n}=\sum _{i=1}^{n}a_{i}X_{i}}</annotation>

</semantics>

</math><img src="./75d9110193a96edf06d72f55351afdf2a3ead412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.236ex; height:3.176ex;" alt="{\textstyle S_{n}=\sum _{i=1}^{n}a_{i}X_{i}}" loading="lazy">, where the Xi are independent random variables and the ai are constants, then the probability density function for Sn is the <a href="Convolution" title="Convolution">convolution</a> of the probability density functions of each of the Xi, and the moment-generating function for Sn is given by

<msub>

<msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{S_{n}}(t)=M_{X_{1}}(a_{1}t)M_{X_{2}}(a_{2}t)\cdots M_{X_{n}}(a_{n}t)\,.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Vector-valued_random_variables">Vector-valued random variables</h3></div>

For <a href="Random_vector" class="mw-redirect" title="Random vector">vector-valued random variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>

</semantics>

</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> with <a href="Real_number" title="Real number">real</a> components, the moment-generating function is given by

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(\mathbf {t} )=\operatorname {E} \left[e^{\langle \mathbf {t} ,\mathbf {X} \rangle }\right]}</annotation>

</semantics>

</math>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} }</annotation>

</semantics>

</math><img src="./16ff63cc74a931900e79b3caacbae3fa8cc66845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} }" loading="lazy"> is a vector and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

</semantics>

</math><img src="./9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> is the <a href="Dot_product" title="Dot product">dot product</a>.

<div class="mw-heading mw-heading2"><h2 id="Important_properties">Important properties</h2></div>

Moment generating functions are positive and <a href="Logarithmically_convex_function" title="Logarithmically convex function">log-convex</a>, with M(0) = 1.

An important property of the moment-generating function is that it uniquely determines the distribution. In other words, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are two random variables and for all values of t,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=M_{Y}(t),}</annotation>

</semantics>

</math>

then

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=F_{Y}(x)}</annotation>

</semantics>

</math>

for all values of x (or equivalently X and Y have the same distribution). This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." This is because in some cases, the moments exist and yet the moment-generating function does not, because the limit

</mrow>

</munder>

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</munderover>

<mfrac>

<mrow>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\sum _{i=0}^{n}{\frac {t^{i}m_{i}}{i!}}}</annotation>

</semantics>

</math>

may not exist. The <a href="Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a> is an example of when this occurs.

<div class="mw-heading mw-heading3"><h3 id="Calculations_of_moments">Calculations of moments</h3></div>

The moment-generating function is so called because if it exists on an open interval around t = 0, then it is the <a href="Exponential_generating_function" class="mw-redirect" title="Exponential generating function">exponential generating function</a> of the <a href="Moment_(mathematics)" title="Moment (mathematics)">moments</a> of the <a href="Probability_distribution" title="Probability distribution">probability distribution</a>:

<msub>

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<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle m_{n}=\operatorname {E} \left[X^{n}\right]=M_{X}^{(n)}(0)=\left.{\frac {d^{n}M_{X}}{dt^{n}}}\right|_{t=0}.}</annotation>

</semantics>

</math>

That is, with n being a nonnegative integer, the n-th moment about 0 is the n-th derivative of the moment generating function, evaluated at t = 0.

<div class="mw-heading mw-heading2"><h2 id="Other_properties">Other properties</h2></div>

<a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a> provides a simple lower bound on the moment-generating function:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)\geq e^{\mu t},}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is the mean of X.

The moment-generating function can be used in conjunction with <a href="Markov's_inequality" title="Markov's inequality">Markov's inequality</a> to bound the upper tail of a real random variable X. This statement is also called the <a href="Chernoff_bound" title="Chernoff bound">Chernoff bound</a>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto e^{xt}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto e^{xt}}</annotation>

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</math><img src="./e31ea78ccc66f8634eea6e78c7799e293c7b9413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.793ex; height:2.509ex;" alt="{\displaystyle x\mapsto e^{xt}}" loading="lazy"> is monotonically increasing for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t>0}</annotation>

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<msup>

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<msup>

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<msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X\geq a)=\Pr(e^{tX}\geq e^{ta})\leq e^{-at}\operatorname {E} \left[e^{tX}\right]=e^{-at}M_{X}(t)}</annotation>

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</math>

for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

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<annotation encoding="application/x-tex">{\displaystyle t>0}</annotation>

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</math><img src="./29a2960e88369263fe3cfe00ccbfeb83daee212a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}" loading="lazy"> and any a, provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)}</annotation>

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</math><img src="./34fcd0a5942fd5fd42e0f8a7d15c301f9851fbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.535ex; height:2.843ex;" alt="{\displaystyle M_{X}(t)}" loading="lazy"> exists. For example, when X is a standard normal distribution and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a>0}</annotation>

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</math><img src="./1f34a80ea013edb56e340b19550430a8b6dfd7b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}" loading="lazy">, we can choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=a}">

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<annotation encoding="application/x-tex">{\displaystyle t=a}</annotation>

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</math><img src="./d5442e0b9221e9617bc41a952450163b67081cda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.168ex; height:2.009ex;" alt="{\displaystyle t=a}" loading="lazy"> and recall that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=e^{t^{2}/2}}">

<msub>

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<msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=e^{t^{2}/2}}</annotation>

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</math><img src="./d3dac56ed8a0829f72fe6588689b7fdc5714f654.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.019ex; height:3.509ex;" alt="{\displaystyle M_{X}(t)=e^{t^{2}/2}}" loading="lazy">. This gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X\geq a)\leq e^{-a^{2}/2}}">

<msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X\geq a)\leq e^{-a^{2}/2}}</annotation>

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</math><img src="./158d20fd6e78e843993ed74de3e3ec0334f7376c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.65ex; height:3.509ex;" alt="{\displaystyle \Pr(X\geq a)\leq e^{-a^{2}/2}}" loading="lazy">, which is within a factor of 1+a of the exact value.

Various lemmas, such as <a href="Hoeffding's_lemma" title="Hoeffding's lemma">Hoeffding's lemma</a> or <a href="Bennett's_inequality" title="Bennett's inequality">Bennett's inequality</a> provide bounds on the moment-generating function in the case of a zero-mean, bounded random variable.

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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<mfrac>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{m}]\leq \left({\frac {m}{te}}\right)^{m}M_{X}(t),}</annotation>

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</math>

For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,m\geq 0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X,m\geq 0}</annotation>

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</math><img src="./b2c1ddcd90099ce6b45d747194a1f181e7586802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.315ex; height:2.509ex;" alt="{\displaystyle X,m\geq 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

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<annotation encoding="application/x-tex">{\displaystyle t>0}</annotation>

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This follows from the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+x\leq e^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle 1+x\leq e^{x}}</annotation>

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</math><img src="./8fc9faef474635a854990cb27002df5dbdf4fea9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.687ex; height:2.509ex;" alt="{\displaystyle 1+x\leq e^{x}}" loading="lazy"> into which we can substitute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'=tx/m-1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x'=tx/m-1}</annotation>

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</math><img src="./40ee020bb3dd93a948f908bf8854c7ed2dc5bdaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.488ex; height:3.009ex;" alt="{\displaystyle x'=tx/m-1}" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tx/m\leq e^{tx/m-1}}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle tx/m\leq e^{tx/m-1}}</annotation>

</semantics>

</math><img src="./a8169c1d94b945fb713494521e01de0ecee52330.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.686ex; height:3.343ex;" alt="{\displaystyle tx/m\leq e^{tx/m-1}}" loading="lazy"> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,t,m\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle x,t,m\in \mathbb {R} }</annotation>

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</math><img src="./5ca687c1f8b8b69ce9b28e7da973bb2a217c8eb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.796ex; height:2.509ex;" alt="{\displaystyle x,t,m\in \mathbb {R} }" loading="lazy">.

Now, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t>0}</annotation>

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</math><img src="./29a2960e88369263fe3cfe00ccbfeb83daee212a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,m\geq 0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x,m\geq 0}</annotation>

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</math><img src="./8df357029ba3c6423e07d2d9ecc10cb0bbae99d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.665ex; height:2.509ex;" alt="{\displaystyle x,m\geq 0}" loading="lazy">, this can be rearranged to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{m}\leq (m/(te))^{m}e^{tx}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{m}\leq (m/(te))^{m}e^{tx}}</annotation>

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</math><img src="./0010e092ac05939e451d0534191fdae77625fb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.372ex; height:3.009ex;" alt="{\displaystyle x^{m}\leq (m/(te))^{m}e^{tx}}" loading="lazy">.

Taking the expectation on both sides gives the bound on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{m}]}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{m}]}</annotation>

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</math><img src="./7d31d9b6f4e262bfe8a57810801515535c37f270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.548ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X^{m}]}" loading="lazy"> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [e^{tX}]}">

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [e^{tX}]}</annotation>

</semantics>

</math><img src="./0379eb85a8f71d1d2e06107ba42758bc26c355b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.186ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [e^{tX}]}" loading="lazy">.

As an example, consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\text{Chi-Squared}}}">

<mtext>Chi-Squared</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\sim {\text{Chi-Squared}}}</annotation>

</semantics>

</math><img src="./d4bb85a83e04e9abbb1280d9ccbc7e3ff06fffc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.682ex; height:2.509ex;" alt="{\displaystyle X\sim {\text{Chi-Squared}}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> degrees of freedom. Then from the <a class="mw-selflink-fragment" href="#Examples">examples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=(1-2t)^{-k/2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=(1-2t)^{-k/2}}</annotation>

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</math><img src="./fd4a7d596cdf9fdbd50d001d976c61d666900734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.459ex; height:3.343ex;" alt="{\displaystyle M_{X}(t)=(1-2t)^{-k/2}}" loading="lazy">.

Picking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=m/(2m+k)}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t=m/(2m+k)}</annotation>

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</math><img src="./a11ab8a0a8a4941ea0da6af90fdcca283655ffe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.205ex; height:2.843ex;" alt="{\displaystyle t=m/(2m+k)}" loading="lazy"> and substituting into the bound:

<msup>

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</msup>

<msup>

<mrow>

<mrow>

</mrow>

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</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{m}]\leq {\left(1+2m/k\right)}^{k/2}e^{-m}{\left(k+2m\right)}^{m}.}</annotation>

</semantics>

</math>

We know that <a href="Chi-square_distribution" class="mw-redirect" title="Chi-square distribution">in this case</a> the correct bound is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{m}]\leq 2^{m}\Gamma (m+k/2)/\Gamma (k/2)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{m}]\leq 2^{m}\Gamma (m+k/2)/\Gamma (k/2)}</annotation>

</semantics>

</math><img src="./49b3e93bf6b119412e575abc6a3af7744b11d2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.124ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X^{m}]\leq 2^{m}\Gamma (m+k/2)/\Gamma (k/2)}" loading="lazy">.

To compare the bounds, we can consider the asymptotics for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

Here the moment-generating function bound is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{m}(1+m^{2}/k+O(1/k^{2}))}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{m}(1+m^{2}/k+O(1/k^{2}))}</annotation>

</semantics>

</math><img src="./814ffe37bdbd8f4e1241c4783a26d2966d9d5a51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.18ex; height:3.176ex;" alt="{\displaystyle k^{m}(1+m^{2}/k+O(1/k^{2}))}" loading="lazy">,

where the real bound is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{m}(1+(m^{2}-m)/k+O(1/k^{2}))}">

<msup>

</mrow>

</msup>

<msup>

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</msup>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{m}(1+(m^{2}-m)/k+O(1/k^{2}))}</annotation>

</semantics>

</math><img src="./e1fdd0395737ba20935eb3a6a4a379173774189b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.87ex; height:3.176ex;" alt="{\displaystyle k^{m}(1+(m^{2}-m)/k+O(1/k^{2}))}" loading="lazy">.

The moment-generating function bound is thus very strong in this case.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_functions">Relation to other functions</h2></div>

Related to the moment-generating function are a number of other <a href="Integral_transform" title="Integral transform">transforms</a> that are common in probability theory:

<dl><dt><a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">Characteristic function</a></dt>

<dd>The <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)}</annotation>

</semantics>

</math><img src="./e2578322e22b80aa79b1e5a4aebf144e5d642c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.801ex; height:2.843ex;" alt="{\displaystyle \varphi _{X}(t)}" loading="lazy"> is related to the moment-generating function via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)=M_{iX}(t)=M_{X}(it):}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)=M_{iX}(t)=M_{X}(it):}</annotation>

</semantics>

</math><img src="./ce4a7fa4964b340512ec6beb044e70d1f719ed64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.731ex; height:2.843ex;" alt="{\displaystyle \varphi _{X}(t)=M_{iX}(t)=M_{X}(it):}" loading="lazy"> the characteristic function is the moment-generating function of iX or the moment generating function of X evaluated on the imaginary axis. This function can also be viewed as the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="Probability_density_function" title="Probability density function">probability density function</a>, which can therefore be deduced from it by inverse Fourier transform.</dd>

<dt><a href="Cumulant-generating_function" class="mw-redirect" title="Cumulant-generating function">Cumulant-generating function</a></dt>

<dd>The <a href="Cumulant-generating_function" class="mw-redirect" title="Cumulant-generating function">cumulant-generating function</a> is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a>, while others call this latter the second cumulant-generating function.</dd>

<dt><a href="Probability-generating_function" title="Probability-generating function">Probability-generating function</a></dt>

<dd>The <a href="Probability-generating_function" title="Probability-generating function">probability-generating function</a> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\operatorname {E} \left[z^{X}\right].}">

<mrow>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)=\operatorname {E} \left[z^{X}\right].}</annotation>

</semantics>

</math><img src="./d52b8167f60d9186498dc3f7b1b0b786a3b2a41f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.714ex; height:3.343ex;" alt="{\displaystyle G(z)=\operatorname {E} \left[z^{X}\right].}" loading="lazy"> This immediately implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(e^{t})=\operatorname {E} \left[e^{tX}\right]=M_{X}(t).}">

<msup>

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle G(e^{t})=\operatorname {E} \left[e^{tX}\right]=M_{X}(t).}</annotation>

</semantics>

</math><img src="./ec22810d622164a684da4b6f970dd0dbd1dbeac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.756ex; height:3.343ex;" alt="{\displaystyle G(e^{t})=\operatorname {E} \left[e^{tX}\right]=M_{X}(t).}" loading="lazy"></dd></dl>

<ul><li><a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">Characteristic function (probability theory)</a></li>

<li><a href="Factorial_moment_generating_function" title="Factorial moment generating function">Factorial moment generating function</a></li>

<li><a href="Rate_function" title="Rate function">Rate function</a></li>

<li><a href="Hamburger_moment_problem" title="Hamburger moment problem">Hamburger moment problem</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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</style><cite id="CITEREFCasellaBerger1990" class="citation book cs1">Casella, George; Berger, Roger L. (1990). Statistical Inference. Wadsworth & Brooks/Cole. p. 61. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-11958-1</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBulmer1979" class="citation book cs1">Bulmer, M. G. (1979). Principles of Statistics. Dover. pp. 75–79. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-63760-3</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Kotz et al. p. 37 using 1 as the number of degree of freedom to recover the Cauchy distribution

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

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<ul><li><cite id="CITEREFCasellaBerger2002" class="citation book cs1">Casella, George; Berger, Roger (2002). Statistical Inference (2nd ed.). Thomson Learning. pp. 59–68. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-24312-8</bdi>.</cite></li></ul>

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</style><div id="Theory_of_probability_distributions39" style="font-size:114%;margin:0 4em">Theory of <a href="Probability_distribution" title="Probability distribution">probability distributions</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Probability_mass_function" title="Probability mass function">probability mass function</a> (pmf)</li>

<li><a href="Probability_density_function" title="Probability density function">probability density function</a> (pdf)</li>

<li><a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (cdf)</li>

<li><a href="Quantile_function" title="Quantile function">quantile function</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Raw_moment" class="mw-redirect" title="Raw moment">raw moment</a></li>

<li><a href="Central_moment" title="Central moment">central moment</a></li>

<li><a href="Variance" title="Variance">variance</a></li>

<li><a href="Standard_deviation" title="Standard deviation">standard deviation</a></li>

<li><a href="Skewness" title="Skewness">skewness</a></li>

<li><a href="Kurtosis" title="Kurtosis">kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moment</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a></li>

<li><a href="Probability-generating_function" title="Probability-generating function">probability-generating function</a> (pgf)</li>

<li><a href="Cumulant" title="Cumulant">cumulant</a></li>

<li><a href="Combinant" title="Combinant">combinant</a></li></ul>

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