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{{Short description|Power series derived from a discrete probability distribution}}
In [[probability theory]], the '''probability generating function''' of a [[discrete random variable]] is a [[power series]] representation (the [[generating function]]) of the [[probability mass function]] of the [[random variable]]. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the [[probability mass function]] for a [[random variable]] ''X'', and to make available the well-developed theory of power series with non-negative coefficients.

==Definition==

=== Univariate case ===
If ''X'' is a [[discrete random variable]] taking values ''x'' in the non-negative [[integer]]s {0,1, ...}, then the ''probability generating function'' of ''X'' is defined as
<ref>{{ cite book | title = Probability and Distribution Theory | author = Gleb Gribakin | url = https://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf }}</ref>

<math display="block">G(z) = \operatorname{E} (z^X) = \sum_{x=0}^{\infty} p(x) z^x,</math>
where <math>p</math> is the [[probability mass function]] of <math>X</math>. Note that the subscripted notations <math>G_X</math> and <math>p_X</math> are often used to emphasize that these pertain to a particular random variable <math>X</math>, and to its [[Probability distribution|distribution]]. The power series [[absolute convergence|converges absolutely]] at least for all [[complex number]]s <math>z</math> with <math>|z|<1</math>; the radius of convergence being often larger.

=== Multivariate case ===
If {{math|1=''X'' = (''X''1,...,''Xd'')}} is a discrete random variable taking values {{math|(''x''1, ..., ''xd'')}} in the {{mvar|d}}-dimensional non-negative [[integer lattice]] {{math|{0,1, ...}''d''}}, then the ''probability generating function'' of {{math|''X''}} is defined as
<math display="block">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},</math>
where {{mvar|p}} is the probability mass function of {{mvar|X}}. The power series converges absolutely at least for all complex vectors <math>z = (z_1, ... z_d) \isin \mathbb{C}^d</math> with <math>\text{max}\{|z_1|, ..., |z_d|\} \le 1.</math>

==Properties==

===Power series===

Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, <math>G(1^-) = 1</math>, where <math>G(1^-) = \lim_{x\to 1, x<1} G(x)</math>, [[One-sided limit|x approaching 1 from below]], since the probabilities must sum to one. So the [[radius of convergence]] of any probability generating function must be at least 1, by [[Abel's theorem]] for power series with non-negative coefficients.

===Probabilities and expectations===

The following properties allow the derivation of various basic quantities related to <math>X</math>:
# The probability mass function of <math>X</math> is recovered by taking [[derivative]]s of <math>G</math>, <math display="block">p(k) = \operatorname{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.</math>
# It follows from Property 1 that if random variables <math>X</math> and <math>Y</math> have probability-generating functions that are equal, <math>G_X = G_Y</math>, then <math>p_X = p_Y</math>. That is, if <math>X</math> and <math>Y</math> have identical probability-generating functions, then they have identical distributions.
# The normalization of the probability mass function can be expressed in terms of the generating function by <math display="block">\operatorname{E}[1] = G(1^-) = \sum_{i=0}^\infty p(i) = 1.</math> The [[expected value|expectation]] of <math>X</math> is given by <math display="block"> \operatorname{E}[X] = G'(1^-).</math> More generally, the <math>k^{th}</math>[[factorial moment]], <math>\operatorname{E}[X(X - 1) \cdots (X - k + 1)]</math> of <math>X</math> is given by <math display="block">\operatorname{E}\left[\frac{X!}{(X-k)!}\right] = G^{(k)}(1^-), \quad k \geq 0.</math> So the [[variance]] of <math>X</math> is given by <math display="block">\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2.</math> Finally, the {{mvar|k}}-th [[raw moment]] of X is given by <math display="block">\operatorname{E}[X^k] = \left(z\frac{\partial}{\partial z}\right)^k G(z) \Big|_{z=1^-}</math>
# <math>G_X(e^t) = M_X(t)</math> where ''X'' is a random variable, <math>G_X(t)</math> is the probability generating function (of <math>X</math>) and <math>M_X(t)</math> is the [[moment-generating function]] (of <math>X</math>).

===Functions of independent random variables===

Probability generating functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example:

{{bullet list
| If <math>X_i, i=1,2,\cdots,N</math> is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
<math display="block">S_N = \sum_{i=1}^N a_i X_i,</math> where the <math>a_i</math> are constant natural numbers, then the probability generating function is given by
<math display="block">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}).</math>
| In particular, if <math>X</math> and <math>Y</math> are independent random variables:
<math display="block">G_{X+Y}(z) = G_X(z) \cdot G_Y(z)</math> and
<math display="block">G_{X-Y}(z) = G_X(z) \cdot G_Y(1/z).</math>
| In the above, the number <math>N</math> of independent random variables in the sequence is fixed. Assume <math>N</math> is discrete random variable taking values on the non-negative integers, which is independent of the <math>X_i</math>, and consider the probability generating function <math>G_N</math>. If the <math>X_i</math> are not only independent but also identically distributed with common probability generating function <math>G_X = G_{X_i}</math>, then
<math display="block">G_{S_N}(z) = G_N(G_X(z)).</math> This can be seen, using the [[law of total expectation]], as follows:
<math display="block">
\begin{align}
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{align}
</math>
This last fact is useful in the study of [[Galton–Watson process]]es and [[compound Poisson process]]es.
| When the <math>X_i</math> are not supposed identically distributed (but still independent and independent of <math>N</math>), we have
<math display="block">G_{S_N}(z) = \sum_{n \ge 1} f_n \prod_{i=1}^n G_{X_i}(z),</math> where <math>f_n = \Pr(N=n).</math> For identically distributed <math>X_i</math>s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of <math>S_N</math> by means of generating functions.
}}

==Examples==

* The probability generating function of an almost surely [[degenerate distribution|constant random variable]], i.e. one with <math>\Pr(X=c) = 1</math> and <math>\Pr(X\neq c) = 0</math> is <math display="block">G(z) = z^c. </math>
* The probability generating function of a [[binomial distribution|binomial random variable]], the number of successes in <math>n</math> trials, with probability <math>p</math> of success in each trial, is <math display="block">G(z) = \left[(1-p) + pz\right]^n. </math> '''Note''': it is the <math>n</math>-fold product of the probability generating function of a [[Bernoulli distribution|Bernoulli random variable]] with parameter <math>p</math>. {{pb}} So the probability generating function of a [[fair coin]], is <math display="block">G(z) = \frac{1}{2} + \frac{z}{2}. </math>
* The probability generating function of a [[negative binomial distribution|negative binomial random variable]] on <math>\{0,1,2 \cdots\}</math>, the number of failures until the <math>r^{th}</math> success with probability of success in each trial <math>p</math>, is <math display="block">G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r,</math> which converges for <math>|z| < \frac{1}{1-p}</math>. {{pb}} '''Note''' that this is the <math>r</math>-fold product of the probability generating function of a [[geometric distribution|geometric random variable]] with parameter <math>1-p</math> on <math>\{0,1,2,\cdots\}</math>.
* The probability generating function of a [[Poisson distribution|Poisson random variable]] with rate parameter <math>\lambda</math> is <math display="block">G(z) = e^{\lambda(z - 1)}.</math>


==Related concepts==

The probability generating function is an example of a [[generating function]] of a sequence: see also [[formal power series]]. It is equivalent to, and sometimes called, the [[z-transform]] of the probability mass function.

Other generating functions of random variables include the [[moment-generating function]], the [[Characteristic function (probability theory)|characteristic function]] and the [[cumulant generating function]]. The probability generating function is also equivalent to the [[factorial moment generating function]], which as <math>\operatorname{E}\left[z^X\right]</math> can also be considered for continuous and other random variables.

{{more citations needed|date=April 2012}}

==Notes==
{{reflist|refs=

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==References==
{{refbegin}}
* {{ cite book | last1 = Johnson | first1 = Norman Lloyd | last2 = Kotz | first2 = Samuel | last3 = Kemp | first3 = Adrienne W. |author3-link=Adrienne W. Kemp| title = Univariate Discrete Distributions | date = 1992 | publisher = J. Wiley & Sons | isbn = 978-0-471-54897-3 | edition = 2nd | series = Wiley series in probability and mathematical statistics | location = New York }}
{{refend}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Probability Generating Function}}
[[Category:Functions related to probability distributions]]
[[Category:Generating functions]]

Probability-generating function - Revision history

imported>Hellacioussatyr at 15:10, 7 August 2025