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'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 === Is Z a constant? define Z! If ''X'' is a [[discrete random variable]] taking values in the non-negative [[integer]]s {0,1, ...}, then the ''probability generating function'' of ''X'' is defined as <ref>http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf</ref> :<math>G(z) = \operatorname{E} (z^X) = \sum_{x=0}^{\infty}p(x)z^x,</math> where ''p'' is the probability mass function of ''X''. Note that the subscripted notations ''G''_''X'' and ''p_X'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its distribution. The power series [[absolute convergence|converges absolutely]] at least for all [[complex number]]s ''z'' with |''z''|&nbsp;≤&nbsp;1; in many examples the radius of convergence is larger. === Multivariate case === If {{nowrap|''X'' {{=}} (''X''₁,...,''X_d''&thinsp;)}} is a discrete random variable taking values in the ''d''-dimensional non-negative [[integer lattice]] {0,1, ...}^''d'', then the ''probability generating function'' of ''X'' is defined as :<math>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 ''p'' is the probability mass function of ''X''. The power series converges absolutely at least for all complex vectors {{nowrap| ''z'' {{=}} (''z''₁,...,''z_d''&thinsp;) ∈ ℂ^''d''}} with {{nowrap|max<nowiki>{|</nowiki>''z''₁<nowiki>|</nowiki>,...,<nowiki>|</nowiki>''z_d''&thinsp;<nowiki>|}</nowiki> ≤ 1}}. ==Properties== ===Power series=== Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, ''G''(1^&minus;) = 1, where ''G''(1^&minus;) = lim_z→1''G''(''z'') [[One-sided limit|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 ''X'': 1. The probability mass function of ''X'' is recovered by taking [[derivative]]s of ''G'' :<math> p(k) = \operatorname{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.</math> 2. It follows from Property 1 that if random variables ''X'' and ''Y'' have probability generating functions that are equal, ''G''_''X'' = ''G''_''Y'', then ''p''_''X'' = ''p''_''Y''. That is, if ''X'' and ''Y'' have identical probability generating functions, then they have identical distributions. 3. The normalization of the probability density function can be expressed in terms of the generating function by :<math>\operatorname{E}(1)=G(1^-)=\sum_{i=0}^\infty f(i)=1.</math> The [[expected value|expectation]] of ''X'' is given by :<math> \operatorname{E}\left(X\right) = G'(1^-).</math> More generally, the ''k''^th [[factorial moment]], <math>\textrm{E}(X(X - 1) \cdots (X - k + 1))</math> of ''X'' is given by :<math>\textrm{E}\left(\frac{X!}{(X-k)!}\right) = G^{(k)}(1^-), \quad k \geq 0.</math> So the [[variance]] of ''X'' is given by :<math>\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2.</math> 4. <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 ''X'') and <math>M_X(t)</math> is the [[moment-generating function]] (of ''X'') . ===Functions of independent random variables=== Probability generating functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example: * If ''X''₁, ''X''₂, ..., ''X''_n is a sequence of independent (and not necessarily identically distributed) random variables, and ::<math>S_n = \sum_{i=1}^n a_i X_i,</math> :where the ''a''_i are constants, then the probability generating function is given by ::<math>G_{S_n}(z) = \operatorname{E}(z^{S_n}) = \operatorname{E}(z^{\sum_{i=1}^n a_i X_i,}) = G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\cdots G_{X_n}(z^{a_n}).</math> :For example, if ::<math>S_n = \sum_{i=1}^n X_i,</math> :then the probability generating function, ''G''_''Sn''(''z''), is given by ::<math>G_{S_n}(z) = G_{X_1}(z)G_{X_2}(z)\cdots G_{X_n}(z).</math> :It also follows that the probability generating function of the difference of two independent random variables ''S'' = ''X''₁ &minus; ''X''₂ is ::<math>G_S(z) = G_{X_1}(z)G_{X_2}(1/z).</math> *Suppose that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function ''G''_''N''. If the ''X''₁, ''X''₂, ..., ''X''_N are independent ''and'' identically distributed with common probability generating function ''G''_X, then ::<math>G_{S_N}(z) = G_N(G_X(z)).</math> :This can be seen, using the [[law of total expectation]], as follows: ::<math> G_{S_N}(z) = \operatorname{E}(z^{S_N}) = \operatorname{E}(z^{\sum_{i=1}^N X_i}) = \operatorname{E}\big(\operatorname{E}(z^{\sum_{i=1}^N X_i}| N) \big) = \operatorname{E}\big( (G_X(z))^N\big) =G_N(G_X(z)).</math> :This last fact is useful in the study of [[Galton&ndash;Watson process]]es. *Suppose again that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function ''G''_''N'' and probability density <math>f_i = \Pr\{N = i\}</math>. If the ''X''₁, ''X''₂, ..., ''X''_N are independent, but ''not'' identically distributed random variables, where <math>G_{X_i}</math> denotes the probability generating function of <math>X_i</math>, then ::<math>G_{S_N}(z) = \sum_{i \ge 1} f_i \prod_{k=1}^i G_{X_i}(z).</math> :For identically distributed ''X_i'' this simplifies to the identity stated before. The general case is sometimes useful to obtain a decomposition of ''S_N'' by means of generating functions. ==Examples== * The probability generating function of a [[degenerate distribution|constant random variable]], i.e. one with Pr(''X'' = ''c'') = 1, is ::<math>G(z) = \left(z^c\right). \, </math> * The probability generating function of a [[binomial distribution|binomial random variable]], the number of successes in ''n'' trials, with probability ''p'' of success in each trial, is ::<math>G(z) = \left[(1-p) + pz\right]^n. \, </math> :Note that this is the ''n''-fold product of the probability generating function of a [[Bernoulli distribution|Bernoulli random variable]] with parameter ''p''. * The probability generating function of a [[negative binomial distribution|negative binomial random variable]] on {0,1,2 ...}, the number of failures until the ''r''th success with probability of success in each trial ''p'', is ::<math>G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r.</math> :(Convergence for <math>|z| < \frac{1}{1-p}</math>). :Note that this is the ''r''-fold product of the probability generating function of a [[geometric distribution|geometric random variable]] with parameter 1−''p'' on {0,1,2 ...}. * The probability generating function of a [[Poisson distribution|Poisson random variable]] with rate parameter λ is ::<math>G(z) = \textrm{e}^{\lambda(z - 1)}.\;\,</math> <!-- TO BE COMPLETED: ==Joint probability generating functions== The concept of the probability generating function for single random variables can be extended to the joint probability generating function of two or more random variables. Suppose that ''X'' and ''Y'' are both discrete random variables (not necessarily independent or identically distributed), again taking values on some subset of the non-negative integers. --> ==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]]. {{refimprove|date=April 2012}} ==Notes== {{reflist|refs= }} ==References== *Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) ''Univariate Discrete distributions'' (2nd edition). Wiley. ISBN 0-471-54897-9 (Section 1.B9) {{Theory of probability distributions}} {{DEFAULTSORT:Probability Generating Function}} [[Category:Theory of probability distributions]] [[Category:Generating functions]]'