ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Probability-generating function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Probability-generating function</span></span>

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<p>In <a href="Probability_theory" title="Probability theory">probability theory</a>, the <b>probability generating function</b> of a <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a> is a <a href="Power_series" title="Power series">power series</a> representation (the <a href="Generating_function" title="Generating function">generating function</a>) of the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> of the <a href="Random_variable" title="Random variable">random variable</a>. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(<i>X</i> = <i>i</i>) in the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> for a <a href="Random_variable" title="Random variable">random variable</a> <i>X</i>, and to make available the well-developed theory of power series with non-negative coefficients.

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

<div class="mw-heading mw-heading3"><h3 id="Univariate_case">Univariate case</h3></div>

<p>If <i>X</i> is a <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a> taking values <i>x</i> in the non-negative <a href="Integer" title="Integer">integers</a> {0,1, ...}, then the <i>probability generating function</i> of <i>X</i> is defined as

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},}">

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<annotation encoding="application/x-tex">{\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> is the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><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"></span>. Note that the subscripted notations <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X}}">

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

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</math></span><img src="./8f059cffb25d6f609ffca5418041a9b378d849e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.459ex; height:2.509ex;" alt="{\displaystyle G_{X}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">

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

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</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> are often used to emphasize that these pertain to a particular random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><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"></span>, and to its <a href="Probability_distribution" title="Probability distribution">distribution</a>. The power series <a href="Absolute_convergence" title="Absolute convergence">converges absolutely</a> at least for all <a href="Complex_number" title="Complex number">complex numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1}">

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<annotation encoding="application/x-tex">{\displaystyle |z|<1}</annotation>

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</math></span><img src="./e1c0fa57b899b653a3823f85f43fd666309c09b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|<1}" loading="lazy"></span>; the radius of convergence being often larger.

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<div class="mw-heading mw-heading3"><h3 id="Multivariate_case">Multivariate case</h3></div>

<p>If <span class="texhtml"><i>X</i> = (<i>X</i><sub>1</sub>,...,<i>X<sub>d</sub></i>)</span> is a discrete random variable taking values <span class="texhtml">(<i>x</i><sub>1</sub>, ..., <i>x<sub>d</sub></i>)</span> in the <span class="texhtml mvar" style="font-style:italic;">d</span>-dimensional non-negative <a href="Integer_lattice" title="Integer lattice">integer lattice</a> <span class="texhtml">{0,1, ...}<sup><i>d</i></sup></span>, then the <i>probability generating function</i> of <span class="texhtml"><i>X</i></span> is defined as

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<annotation encoding="application/x-tex">{\displaystyle G(z)=G(z_{1},\ldots ,z_{d})=\operatorname {E} {\bigl (}z_{1}^{X_{1}}\cdots z_{d}^{X_{d}}{\bigr )}=\sum _{x_{1},\ldots ,x_{d}=0}^{\infty }p(x_{1},\ldots ,x_{d})z_{1}^{x_{1}}\cdots z_{d}^{x_{d}},}</annotation>

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where <span class="texhtml mvar" style="font-style:italic;">p</span> is the probability mass function of <span class="texhtml mvar" style="font-style:italic;">X</span>. The power series converges absolutely at least for all complex vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=(z_{1},...z_{d})\in \mathbb {C} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle z=(z_{1},...z_{d})\in \mathbb {C} ^{d}}</annotation>

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</math></span><img src="./b4e62838bf9687b96f700cd04565cf23f92e59a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.051ex; height:3.176ex;" alt="{\displaystyle z=(z_{1},...z_{d})\in \mathbb {C} ^{d}}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{max}}\{|z_{1}|,...,|z_{d}|\}\leq 1.}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{max}}\{|z_{1}|,...,|z_{d}|\}\leq 1.}</annotation>

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</math></span><img src="./cbd0d9c244aa4de22c1efc4fe7dcb5a695fca1ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.624ex; height:2.843ex;" alt="{\displaystyle {\text{max}}\{|z_{1}|,...,|z_{d}|\}\leq 1.}" loading="lazy"></span>

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

<div class="mw-heading mw-heading3"><h3 id="Power_series">Power series</h3></div>

<p>Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(1^{-})=1}">

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

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</math></span><img src="./3fa5571ec30a1e911340955227c35b7234c89e11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.57ex; height:3.009ex;" alt="{\displaystyle G(1^{-})=1}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(1^{-})=\lim _{x\to 1,x<1}G(x)}">

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<annotation encoding="application/x-tex">{\displaystyle G(1^{-})=\lim _{x\to 1,x<1}G(x)}</annotation>

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</math></span><img src="./842ba7bb903559851441b88c25b6384af7986e21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.664ex; height:4.509ex;" alt="{\displaystyle G(1^{-})=\lim _{x\to 1,x<1}G(x)}" loading="lazy"></span>, <a href="One-sided_limit" title="One-sided limit">x approaching 1 from below</a>, since the probabilities must sum to one. So the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of any probability generating function must be at least 1, by <a href="Abel's_theorem" title="Abel's theorem">Abel's theorem</a> for power series with non-negative coefficients.

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<div class="mw-heading mw-heading3"><h3 id="Probabilities_and_expectations">Probabilities and expectations</h3></div>

<p>The following properties allow the derivation of various basic quantities related to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><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"></span>:

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<ol><li>The probability mass function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><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"></span> is recovered by taking <a href="Derivative" title="Derivative">derivatives</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k)=\operatorname {Pr} (X=k)={\frac {G^{(k)}(0)}{k!}}.}">

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<annotation encoding="application/x-tex">{\displaystyle p(k)=\operatorname {Pr} (X=k)={\frac {G^{(k)}(0)}{k!}}.}</annotation>

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<li>It follows from Property 1 that if random variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math></span><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"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math></span><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"></span> have probability-generating functions that are equal, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X}=G_{Y}}">

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

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</math></span><img src="./131118e3207a6eb87f1c5f40fae6949f1979ecfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.87ex; height:2.509ex;" alt="{\displaystyle G_{X}=G_{Y}}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}=p_{Y}}">

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

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</math></span><img src="./eef04d80fd6fec192101c86c0f6902c71f8ce655.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.645ex; height:2.009ex;" alt="{\displaystyle p_{X}=p_{Y}}" loading="lazy"></span>. That is, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><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"></span> have identical probability-generating functions, then they have identical distributions.</li>

<li>The normalization of the probability mass function can be expressed in terms of the generating function by <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [1]=G(1^{-})=\sum _{i=0}^{\infty }p(i)=1.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [1]=G(1^{-})=\sum _{i=0}^{\infty }p(i)=1.}</annotation>

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</math></span></span> The <a href="Expected_value" title="Expected value">expectation</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math></span><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"></span> is given by <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]=G'(1^{-}).}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=G'(1^{-}).}</annotation>

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</math></span></span> More generally, the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{th}}">

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</math></span><img src="./c348e1c6f8200f15d1d6026fc140d554b272096d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.984ex; height:2.676ex;" alt="{\displaystyle k^{th}}" loading="lazy"></span><a href="Factorial_moment" title="Factorial moment">factorial moment</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X(X-1)\cdots (X-k+1)]}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X(X-1)\cdots (X-k+1)]}</annotation>

</semantics>

</math></span><img src="./af160eb8d9644bbb51777d130dd2fd5211336760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.99ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X(X-1)\cdots (X-k+1)]}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is given by <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[{\frac {X!}{(X-k)!}}\right]=G^{(k)}(1^{-}),\quad k\geq 0.}">

<mrow>

<mfrac>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[{\frac {X!}{(X-k)!}}\right]=G^{(k)}(1^{-}),\quad k\geq 0.}</annotation>

</semantics>

</math></span></span> So the <a href="Variance" title="Variance">variance</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><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"></span> is given by <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (X)=G''(1^{-})+G'(1^{-})-\left[G'(1^{-})\right]^{2}.}">

<msup>

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</msup>

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</msup>

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</msup>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X)=G''(1^{-})+G'(1^{-})-\left[G'(1^{-})\right]^{2}.}</annotation>

</semantics>

</math></span></span> Finally, the <span class="texhtml mvar" style="font-style:italic;">k</span>-th <a href="Raw_moment" class="mw-redirect" title="Raw moment">raw moment</a> of X is given by <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{k}]=\left(z{\frac {\partial }{\partial z}}\right)^{k}G(z){\Big |}_{z=1^{-}}}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{k}]=\left(z{\frac {\partial }{\partial z}}\right)^{k}G(z){\Big |}_{z=1^{-}}}</annotation>

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</math></span></span></li>

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

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</math></span><img src="./909baa56aa2cb962d6e2d4d6974576f38da2a67b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.812ex; height:3.009ex;" alt="{\displaystyle G_{X}(e^{t})=M_{X}(t)}" loading="lazy"></span> where <i>X</i> is a random variable, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X}(t)}">

<msub>

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

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</math></span><img src="./709762c9262642c29e530ad0e9eccb0c82e24a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:2.843ex;" alt="{\displaystyle G_{X}(t)}" loading="lazy"></span> is the probability generating function (of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>) and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)}">

<msub>

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

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</math></span><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"></span> is the <a href="Moment-generating_function" title="Moment-generating function">moment-generating function</a> (of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>).</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Functions_of_independent_random_variables">Functions of independent random variables</h3></div>

<p>Probability generating functions are particularly useful for dealing with functions of <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> random variables. For example:

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle X_{i},i=1,2,\cdots ,N}</annotation>

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</math></span><img src="./b4dda59cd7e2be52fe50bf41027250686a441344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.259ex; height:2.509ex;" alt="{\displaystyle X_{i},i=1,2,\cdots ,N}" loading="lazy"></span> is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and

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</munderover>

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

</semantics>

</math></span></span> where the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

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

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</math></span><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> are constant natural numbers, then the probability generating function is given by

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle G_{S_{N}}(z)=\operatorname {E} (z^{S_{N}})=\operatorname {E} \left(z^{\sum _{i=1}^{N}a_{i}X_{i},}\right)=G_{X_{1}}(z^{a_{1}})G_{X_{2}}(z^{a_{2}})\cdots G_{X_{N}}(z^{a_{N}}).}</annotation>

</semantics>

</math></span></span></li><li>In particular, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> are independent random variables:

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<annotation encoding="application/x-tex">{\displaystyle G_{X+Y}(z)=G_{X}(z)\cdot G_{Y}(z)}</annotation>

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</math></span></span> and

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<annotation encoding="application/x-tex">{\displaystyle G_{X-Y}(z)=G_{X}(z)\cdot G_{Y}(1/z).}</annotation>

</semantics>

</math></span></span></li><li>In the above, the number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

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

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> of independent random variables in the sequence is fixed. Assume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is discrete random variable taking values on the non-negative integers, which is independent of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

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

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</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span>, and consider the probability generating function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

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

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</math></span><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"></span>. If the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

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</mrow>

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

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> are not only independent but also identically distributed with common probability generating function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X}=G_{X_{i}}}">

<msub>

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<msub>

<msub>

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

</semantics>

</math></span><img src="./136e51dfc7327d16e912f77ff82fdf03f20ee66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.602ex; height:2.843ex;" alt="{\displaystyle G_{X}=G_{X_{i}}}" loading="lazy"></span>, then

<msub>

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<annotation encoding="application/x-tex">{\displaystyle G_{S_{N}}(z)=G_{N}(G_{X}(z)).}</annotation>

</semantics>

</math></span></span> This can be seen, using the <a href="Law_of_total_expectation" title="Law of total expectation">law of total expectation</a>, as follows:

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G_{S_{N}}(z)&=\operatorname {E} (z^{S_{N}})=\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}})\\[4pt]&=\operatorname {E} {\big (}\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}}\mid N){\big )}=\operatorname {E} {\big (}(G_{X}(z))^{N}{\big )}=G_{N}(G_{X}(z)).\end{aligned}}}</annotation>

</semantics>

</math></span></span>

This last fact is useful in the study of <a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson processes</a> and <a href="Compound_Poisson_process" title="Compound Poisson process">compound Poisson processes</a>.</li><li>When the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

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

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> are not supposed identically distributed (but still independent and independent of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>), we have

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<annotation encoding="application/x-tex">{\displaystyle G_{S_{N}}(z)=\sum _{n\geq 1}f_{n}\prod _{i=1}^{n}G_{X_{i}}(z),}</annotation>

</semantics>

</math></span></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}=\Pr(N=n).}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}=\Pr(N=n).}</annotation>

</semantics>

</math></span><img src="./1e2ab60a97bdc3511b15205d85dce351467e2228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.964ex; height:2.843ex;" alt="{\displaystyle f_{n}=\Pr(N=n).}" loading="lazy"></span> For identically distributed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

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

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span>s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./aca805f5a6c6548a825d119230ab751152321ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.116ex; height:2.509ex;" alt="{\displaystyle S_{N}}" loading="lazy"></span> by means of generating functions.</li></ul></div>

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

<ul><li>The probability generating function of an almost surely <a href="Degenerate_distribution" title="Degenerate distribution">constant random variable</a>, i.e. one with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X=c)=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2e50853f385fbbce6092170849a8a009114c87d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.65ex; height:2.843ex;" alt="{\displaystyle \Pr(X=c)=1}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X\neq c)=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d1b279633dd570f376cbc4975596cdf67d1293f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.65ex; height:2.843ex;" alt="{\displaystyle \Pr(X\neq c)=0}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=z^{c}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)=z^{c}.}</annotation>

</semantics>

</math></span></span></li>

<li>The probability generating function of a <a href="Binomial_distribution" title="Binomial distribution">binomial random variable</a>, the number of successes in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> trials, with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> of success in each trial, is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\left[(1-p)+pz\right]^{n}.}">

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)=\left[(1-p)+pz\right]^{n}.}</annotation>

</semantics>

</math></span></span> <b>Note</b>: it is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-fold product of the probability generating function of a <a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli random variable</a> with parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>. <div class="paragraphbreak" style="margin-top:0.5em"></div> So the probability generating function of a <a href="Fair_coin" title="Fair coin">fair coin</a>, is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)={\frac {1}{2}}+{\frac {z}{2}}.}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)={\frac {1}{2}}+{\frac {z}{2}}.}</annotation>

</semantics>

</math></span></span></li>

<li>The probability generating function of a <a href="Negative_binomial_distribution" title="Negative binomial distribution">negative binomial random variable</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2\cdots \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,1,2\cdots \}}</annotation>

</semantics>

</math></span><img src="./f1155f53f52527a804e242ad2e2edd65fe97cf53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.99ex; height:2.843ex;" alt="{\displaystyle \{0,1,2\cdots \}}" loading="lazy"></span>, the number of failures until the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{th}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f1095b65e0b65d596b93fbc5bbc6732fa617af89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.821ex; height:2.676ex;" alt="{\displaystyle r^{th}}" loading="lazy"></span> success with probability of success in each trial <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>, is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\left({\frac {p}{1-(1-p)z}}\right)^{r},}">

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)=\left({\frac {p}{1-(1-p)z}}\right)^{r},}</annotation>

</semantics>

</math></span></span> which converges for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<{\frac {1}{1-p}}}">

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |z|<{\frac {1}{1-p}}}</annotation>

</semantics>

</math></span><img src="./5b43db701a3f974282baafe9f85941854521adf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.489ex; height:5.676ex;" alt="{\displaystyle |z|<{\frac {1}{1-p}}}" loading="lazy"></span>. <div class="paragraphbreak" style="margin-top:0.5em"></div> <b>Note</b> that this is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>-fold product of the probability generating function of a <a href="Geometric_distribution" title="Geometric distribution">geometric random variable</a> with parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\cdots \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\cdots \}}</annotation>

</semantics>

</math></span><img src="./28b62888d9b6c58ea8924aa9546ce1589b4bcc46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.637ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\cdots \}}" loading="lazy"></span>.</li>

<li>The probability generating function of a <a href="Poisson_distribution" title="Poisson distribution">Poisson random variable</a> with rate parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=e^{\lambda (z-1)}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2></div>

<p>The probability generating function is an example of a <a href="Generating_function" title="Generating function">generating function</a> of a sequence: see also <a href="Formal_power_series" title="Formal power series">formal power series</a>. It is equivalent to, and sometimes called, the <a href="Z-transform" title="Z-transform">z-transform</a> of the probability mass function.

</p><p>Other generating functions of random variables include the <a href="Moment-generating_function" title="Moment-generating function">moment-generating function</a>, the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> and the <a href="Cumulant_generating_function" class="mw-redirect" title="Cumulant generating function">cumulant generating function</a>. The probability generating function is also equivalent to the <a href="Factorial_moment_generating_function" title="Factorial moment generating function">factorial moment generating function</a>, which as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[z^{X}\right]}">

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3bf9c3fd603e9f2eea52671e6af913e4ddb6276f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.245ex; height:3.343ex;" alt="{\displaystyle \operatorname {E} \left[z^{X}\right]}" loading="lazy"></span> can also be considered for continuous and other random variables.

</p>

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

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/* start https://en.wikipedia.org/ */

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

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</style><cite id="CITEREFGleb_Gribakin" class="citation book cs1">Gleb Gribakin. <a rel="nofollow" class="external text" href="https://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf"><i>Probability and Distribution Theory</i></a> <span class="cs1-format">(PDF)</span>.</cite></span>

</li>

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

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

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<ul><li><cite id="CITEREFJohnsonKotzKemp1992" class="citation book cs1">Johnson, Norman Lloyd; Kotz, Samuel; <a href="Adrienne_W._Kemp" title="Adrienne W. Kemp">Kemp, Adrienne W.</a> (1992). <i>Univariate Discrete Distributions</i>. Wiley series in probability and mathematical statistics (2nd ed.). New York: J. Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-54897-3</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><span typeof="mw:File"></span></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">

<ul><li><a href="Moment-generating_function" title="Moment-generating function">moment-generating function</a> (mgf)</li>

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

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

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

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