Inprobability theory, theprobability generating function of adiscrete random variable is apower series representation (thegenerating function) of theprobability mass function of therandom variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X =i) in theprobability mass function for arandom variableX, and to make available the well-developed theory of power series with non-negative coefficients.

IfX is adiscrete random variable taking valuesx in the non-negativeintegers {0,1, ...}, then theprobability generating function ofX is defined as^[1]

$${\displaystyle G(z)=\mathrm{E}(z^X)=\sum _{x=0}^{\mathrm{\infty }}p(x)z^x,}$$where${\displaystyle p}$ is theprobability mass function of${\displaystyle X}$. Note that the subscripted notations${\displaystyle G_X}$ and${\displaystyle p_X}$ are often used to emphasize that these pertain to a particular random variable${\displaystyle X}$, and to itsdistribution. The power seriesconverges absolutely at least for allcomplex numbers${\displaystyle z}$ with; the radius of convergence being often larger.

IfX = (X₁,...,X_d) is a discrete random variable taking values(x₁, ...,x_d) in thed-dimensional non-negativeinteger lattice{0,1, ...}^d, then theprobability generating function ofX is defined as$${\displaystyle G(z)=G(z_1,\dots ,z_d)=\mathrm{E}(z_1^{X_1}\cdots z_d^{X_d})=\sum _{x_1,\dots ,x_d=0}^{\mathrm{\infty }}p(x_1,\dots ,x_d)z_1^{x_1}\cdots z_d^{x_d},}$$wherep is the probability mass function ofX. The power series converges absolutely at least for all complex vectors${\displaystyle z=(z_1,...z_d)\in {\mathbb{C}}^d}$ with${\displaystyle \text{max}\{|z_1|,...,|z_d|\}\le 1.}$

Probability generating functions obey all the rules of power series with non-negative coefficients. In particular,${\displaystyle G(1^{-})=1}$, where,x approaching 1 from below, since the probabilities must sum to one. So theradius of convergence of any probability generating function must be at least 1, byAbel's theorem for power series with non-negative coefficients.

The following properties allow the derivation of various basic quantities related to${\displaystyle X}$:

1. The probability mass function of${\displaystyle X}$ is recovered by takingderivatives of${\displaystyle G}$,$${\displaystyle p(k)=\mathrm{Pr}(X=k)=\frac{G^{(k)}(0)}{k!}.}$$
2. It follows from Property 1 that if random variables${\displaystyle X}$ and${\displaystyle Y}$ have probability-generating functions that are equal,${\displaystyle G_X=G_Y}$, then${\displaystyle p_X=p_Y}$. That is, if${\displaystyle X}$ and${\displaystyle Y}$ have identical probability-generating functions, then they have identical distributions.
3. The normalization of the probability mass function can be expressed in terms of the generating function by$${\displaystyle \mathrm{E}[1]=G(1^{-})=\sum _{i=0}^{\mathrm{\infty }}p(i)=1.}$$ Theexpectation of${\displaystyle X}$ is given by$${\displaystyle \mathrm{E}[X]=G^{\prime }(1^{-}).}$$ More generally, the${\displaystyle k^{th}}$factorial moment,${\displaystyle \mathrm{E}[X(X-1)\cdots (X-k+1)]}$ of${\displaystyle X}$ is given by$${\displaystyle \mathrm{E}\left[\frac{X!}{(X-k)!}\right]=G^{(k)}(1^{-}),k\ge 0.}$$ So thevariance of${\displaystyle X}$ is given by$${\displaystyle \mathrm{Var}(X)=G^{″}(1^{-})+G^{\prime }(1^{-})-{\left[G^{\prime }(1^{-})\right]}^2.}$$ Finally, thek-thraw moment of X is given by$${\displaystyle \mathrm{E}[X^k]={\left(z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}^{k}G(z){|}_{z=1^{-}}}$$
4. ${\displaystyle G_X(e^t)=M_X(t)}$ whereX is a random variable,${\displaystyle G_X(t)}$ is the probability generating function (of${\displaystyle X}$) and${\displaystyle M_X(t)}$ is themoment-generating function (of${\displaystyle X}$).

Probability generating functions are particularly useful for dealing with functions ofindependent random variables. For example:

• If${\displaystyle X_i,i=1,2,\cdots ,N}$ is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and

$${\displaystyle S_N=\sum _{i=1}^N{a}_i{X}_i,}$$ where the${\displaystyle a_i}$ are constant natural numbers, then the probability generating function is given by$${\displaystyle G_{S_N}(z)=\mathrm{E}(z^{S_N})=\mathrm{E}\left(z^{\sum _{i=1}^N{a}_i{X}_i,}\right)=G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\cdots G_{X_N}(z^{a_N}).}$$

• In particular, if${\displaystyle X}$ and${\displaystyle Y}$ are independent random variables:

$${\displaystyle G_{X+Y}(z)=G_X(z)\cdot G_Y(z)}$$ and$${\displaystyle G_{X-Y}(z)=G_X(z)\cdot G_Y(1/z).}$$

• In the above, the number${\displaystyle N}$ of independent random variables in the sequence is fixed. Assume${\displaystyle N}$ is discrete random variable taking values on the non-negative integers, which is independent of the${\displaystyle X_i}$, and consider the probability generating function${\displaystyle G_N}$. If the${\displaystyle X_i}$ are not only independent but also identically distributed with common probability generating function${\displaystyle G_X=G_{X_i}}$, then

$${\displaystyle G_{S_N}(z)=G_N(G_X(z)).}$$ This can be seen, using thelaw of total expectation, as follows:This last fact is useful in the study ofGalton–Watson processes andcompound Poisson processes.

• When the${\displaystyle X_i}$ are not supposed identically distributed (but still independent and independent of${\displaystyle N}$), we have

$${\displaystyle G_{S_N}(z)=\sum _{n\ge 1}{f}_n\prod _{i=1}^n{G}_{X_i}(z),}$$ where${\displaystyle f_n=Pr(N=n).}$ For identically distributed${\displaystyle X_i}$s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of${\displaystyle S_N}$ by means of generating functions.

• The probability generating function of an almost surelyconstant random variable, i.e. one with${\displaystyle Pr(X=c)=1}$ and${\displaystyle Pr(X\ne c)=0}$ is$${\displaystyle G(z)=z^c.}$$
• The probability generating function of abinomial random variable, the number of successes in${\displaystyle n}$ trials, with probability${\displaystyle p}$ of success in each trial, is$${\displaystyle G(z)={\left[(1-p)+pz\right]}^n.}$$Note: it is the${\displaystyle n}$-fold product of the probability generating function of aBernoulli random variable with parameter${\displaystyle p}$.
  So the probability generating function of afair coin, is$${\displaystyle G(z)=1/2+z/2.}$$
• The probability generating function of anegative binomial random variable on${\displaystyle \{0,1,2\cdots \}}$, the number of failures until the${\displaystyle r^{th}}$ success with probability of success in each trial${\displaystyle p}$, is$${\displaystyle G(z)={\left(\frac{p}{1-(1-p)z}\right)}^r,}$$ which converges for.
  Note that this is the${\displaystyle r}$-fold product of the probability generating function of ageometric random variable with parameter${\displaystyle 1-p}$ on${\displaystyle \{0,1,2,\cdots \}}$.
• The probability generating function of aPoisson random variable with rate parameter${\displaystyle \lambda }$ is$${\displaystyle G(z)=e^{\lambda (z-1)}.}$$

The probability generating function is an example of agenerating function of a sequence: see alsoformal power series. It is equivalent to, and sometimes called, thez-transform of the probability mass function.

Other generating functions of random variables include themoment-generating function, thecharacteristic function and thecumulant generating function. The probability generating function is also equivalent to thefactorial moment generating function, which as${\displaystyle \mathrm{E}\left[z^X\right]}$ can also be considered for continuous and other random variables.

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Probability-generating function" – news ·newspapers ·books ·scholar ·JSTOR(April 2012) (Learn how and when to remove this message)

• Johnson, Norman Lloyd; Kotz, Samuel;Kemp, Adrienne W. (1992).Univariate Discrete Distributions. Wiley series in probability and mathematical statistics (2nd ed.). New York: J. Wiley & Sons.ISBN 978-0-471-54897-3.

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