# Probability-generating function

A probability-generating function (PGF) is a power series that encodes the probability mass function of a nonnegative integer-valued random variable: for a random variable X taking values 0, 1, 2, ... with probabilities p_n = P(X = n), the PGF is

\[ G_X(s) = \mathbb{E}[s^X] = \sum_{n=0}^{\infty} p_n s^n. \]

The name comes from the fact that the probabilities can be recovered from the function: expanding G_X(s) as a power series in s, the coefficient of s^n is p_n.<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> PGFs are a major analytical tool for stochastic processes on discrete state spaces, such as branching processes and queueing systems.<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( G_X(s) = \mathbb{E}[s^X] = \sum_{n=0}^{\infty} p_n s^n \) for a nonnegative integer-valued random variable<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> |
| Recovering probabilities | \( p_k = \frac{1}{k!} G_X^{(k)}(0) \), so the PGF uniquely determines the distribution<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> |
| Mean and variance | \( \mathbb{E}(X) = G'_X(1) \) and \( \mathrm{Var}(X) = G''_X(1) + G'_X(1) - (G'_X(1))^2 \)<sup>[2](https://pastopic.com/cambridge/further-mathematics-9231/notes/probability-generating-functions)</sup> |
| Convergence | The radius of convergence R exists for every PGF and satisfies \( R \ge 1 \); it can be anything from R = 1 to R = \( \infty \)<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup> |
| Sums of variables | For independent X and Y, \( G_{X+Y}(s) = G_X(s)\,G_Y(s) \); for a random sum \( Z_N = X_1 + \cdots + X_N \), the PGF is \( h(g(s)) \)<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup><sup> • </sup><sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>  |
| Branching processes | The extinction probability q of a Galton-Watson process is the smallest nonnegative fixed point of the offspring PGF, \( G_Y(q) = q \)<sup>[4](https://link.springer.com/article/10.1007/s13324-026-01200-y)</sup> |
| History | The technique appears in de Moivre's *Miscellanea Analytica* (1730); Laplace named and developed it in 1812; the English term dates only from the 1940s<sup>[5](https://mathshistory.st-andrews.ac.uk/Miller/mathword/p/)</sup> |

## Basic properties

Because a PGF is a power series with nonnegative coefficients summing to 1, it converges at s = 1 and is infinitely differentiable inside its [radius of convergence](https://www.edgepedia.org/radius-of-convergence). Differentiating at s = 0 recovers the mass function term by term: \( p_k = \frac{1}{k!} G_X^{(k)}(0) \). An immediate consequence is uniqueness: if two nonnegative integer-valued random variables X and Y have PGFs that agree for all s in some interval containing 0, then X and Y have the same distribution.<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> This licenses the standard argument that a generating function identified as that of, say, a [Poisson distribution](https://www.edgepedia.org/poisson-distribution) proves the variable is Poisson.<sup>[6](https://wjmaths.com/courses/probability-theory/build/probability_theorysu12.html)</sup>

**Moments from derivatives at s = 1.** The factorial moments of X are generated by derivatives at s = 1: \( \mathbb{E}[X(X-1)\cdots(X-k+1)] = G_X^{(k)}(1) \), which exists and is finite exactly when the corresponding derivative does.<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> From this follow the working formulas \( \mathbb{E}(X) = G'_X(1) \) and \( \mathrm{Var}(X) = G''_X(1) + G'_X(1) - (G'_X(1))^2 \).<sup>[2](https://pastopic.com/cambridge/further-mathematics-9231/notes/probability-generating-functions)</sup> Equivalently, the rth moment can be extracted with the operator \( x\,\partial_x \) applied to the PGF and evaluated at x = 1.<sup>[7](https://joel-miller-lab.github.io/AMSI2026_PGF/notebooks/BasicGenFxn/GenFxnsDiff.html)</sup>

**Examples.** For X ~ Bernoulli(p), \( G_X(s) = (1-p) + ps \). For X ~ Poisson(λ), \( G_X(s) = e^{-\lambda(1-s)} \), whose derivatives at s = 1 all equal powers of λ, giving \( \mathbb{E}(X) = \lambda \).<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup> For X ~ Binomial(n, p), \( G_X(s) = (ps + q)^n \) with q = 1 - p.<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>

## Convergence and the radius of convergence

The radius of convergence of a PGF is a number R > 0 such that the sum \( G_X(s) = \sum_{x=0}^{\infty} s^x \mathbb{P}(X=x) \) converges if \( |s| < R \) and diverges if \( |s| > R \); no general statement is made at \( |s| = R \).<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup> For any PGF the radius exists, is always at least 1, and can be anything from R = 1 to R = \( \infty \). The set of s for which the series converges is symmetric about 0, even though the function itself generally is not, since \( G_X(-s) \neq G_X(s) \) in general.<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>

The two standard examples illustrate the range. For X ~ Geometric(p = 0.8), \( G_X(s) = 0.8/(1 - 0.2s) \), valid for \( |0.2s| < 1 \), so the radius is R = 5. For X ~ Binomial(n, p), the sum is finite and \( G_X(s) = (ps+q)^n \) holds for all real s, so R = \( \infty \).<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>

**Abel's theorem.** Behavior at the edge of convergence can differ at the two ends: for the Geometric(0.8) example the PGF is not continuous at s = -5, but Abel's theorem states that a power series with nonnegative coefficients is always left-continuous at s = 1, whether or not the sum there is finite. Since R ≥ 1 for every PGF, this guarantees continuity at s = 1 even in the worst case R = 1, and it justifies computing \( \mathbb{E}(X) = G'_X(1) \) by taking the limit of \( G'_X(s) \) as s rises to 1.<sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>

## Sums and compound distributions

PGFs convert convolution into multiplication. For independent X and Y, \( G_{X+Y}(s) = G_X(s) G_Y(s) \), and for n i.i.d. variables with common PGF g(s), the sum has PGF \( [g(s)]^n \).<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup><sup> • </sup><sup>[3](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)</sup>  This is how the PGF identifies distributions of sums: adding two independent Poisson(λ) and Poisson(µ) variables gives the PGF of a Poisson(λ + µ) variable, and uniqueness completes the proof.<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup>

For a compound process, a random sum \( Z_N = X_1 + \cdots + X_N \) where N is independent of the i.i.d. summands, the PGF is the composition \( G(s) = h(g(s)) \), with h the PGF of N and g the common PGF of the summands. Differentiating gives \( \mathbb{E}[Z_N] = \mathbb{E}[N]\,\mathbb{E}[X_1] \).<sup>[1](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)</sup>

## Applications

**Branching processes.** In a Galton-Watson process, where every individual independently produces a random number of children from an offspring distribution Y, the generating function \( f(s) = \mathbb{E}[s^{Z_1}] \) of one generation determines the process: the population of the next generation is a sum of i.i.d. copies, so the transition probabilities have generating function \( [f(s)]^i \) from state i.<sup>[8](https://math.uchicago.edu/~may/REU2024/REUPapers/Singh.pdf)</sup> A classical result is that the extinction probability q is the smallest nonnegative fixed point of the offspring PGF, \( G_Y(q) = q \); the first complete proof was given by J. F. Steffensen in a 1930 note written in Danish.<sup>[4](https://link.springer.com/article/10.1007/s13324-026-01200-y)</sup> The same composition property underlies the recurrence relation for the PGF of the population size in the nth generation.<sup>[9](https://ideas.repec.org/h/spr/sprchp/978-981-99-5601-2_5.html)</sup>

**Queueing and epidemics.** In the M/G/1 queueing system, the PGF of the occupancy distribution is developed first and then used to obtain Laplace-Stieltjes transforms for the waiting-time, sojourn-time, and busy-period distributions.<sup>[10](https://ndl.ethernet.edu.et/bitstream/123456789/23430/1/273.pdf.pdf)</sup> In epidemic modeling, PGFs of offspring (transmission) distributions predict early outbreak behavior in large populations, including the probability of an epidemic, the size distribution after a given number of generations, and the cumulative size distribution of non-epidemic outbreaks.<sup>[11](https://arxiv.org/abs/1803.05136)</sup>

## Relation to other generating functions

The PGF is one of four generating functions commonly used in statistics, alongside the moment-generating function (MGF), the cumulant generating function, and the characteristic function.<sup>[12](https://onlinelibrary.wiley.com/doi/10.1002/9781119047063.ch9)</sup> The MGF is \( M_X(t) = \mathbb{E}[e^{tX}] \) and generates ordinary moments through \( \mathbb{E}(X^j) = M_X^{(j)}(0) \), but it may not exist; the characteristic function \( \phi_X(t) = \mathbb{E}[e^{itX}] \) always exists and uniquely determines the distribution.<sup>[13](https://faculty.washington.edu/yenchic/26A_stat516/Lec1_probability.html)</sup><sup> • </sup><sup>[14](https://r-statistics.co/Moment-Generating-Functions-vs-Characteristic-Functions.html)</sup>  The PGF applies to nonnegative integer-valued variables, where it is essentially the MGF evaluated at \( t = \ln s \), and its derivatives at zero return the probability mass function itself rather than the moments.<sup>[15](https://www.sciencedirect.com/topics/mathematics/probability-generating-function)</sup>

## History

Abraham de Moivre used the generating-function technique to count the chances of throwing s points with n dice in his *Miscellanea Analytica* (1730), with the analysis reproduced in the second edition of the *Doctrine of Chances* (1738); Thomas Simpson also used generating functions in 1755. The method owes its name to [Pierre-Simon Laplace](https://www.edgepedia.org/pierre-simon-laplace), who also advanced it; in the 1812 first book of his *Théorie Analytique des Probabilités*, entitled *Calcul des Functions Génératrices*, he treats it. The specific English term probability generating function has been current only since the 1940s, with an early appearance in [M. S. Bartlett](https://www.edgepedia.org/m-s-bartlett)'s 1940 paper in the *Journal of the Royal Statistical Society*.<sup>[5](https://mathshistory.st-andrews.ac.uk/Miller/mathword/p/)</sup>

## References

1. [Probability Generating Functions, University of Florida lecture notes](https://users.phhp.ufl.edu/rlp176/Courses/BasicAnalysis/PGF.pdf)
2. [Probability generating functions notes, Cambridge Further Mathematics 9231, Pastopic](https://pastopic.com/cambridge/further-mathematics-9231/notes/probability-generating-functions)
3. [Chapter 4: Probability Generating Functions, Stat 325, University of Auckland](https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf)
4. [Galton-Watson processes, simple varieties of trees and Khinchin families, Analysis and Mathematical Physics (Springer)](https://link.springer.com/article/10.1007/s13324-026-01200-y)
5. [Earliest Known Uses of Some of the Words of Mathematics (P), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathword/p/)
6. [Uniqueness and Continuity, Probability Theory course notes](https://wjmaths.com/courses/probability-theory/build/probability_theorysu12.html)
7. [Differentiation of Generating Functions, Applications of PGFs to Biological Systems, Joel Miller Lab](https://joel-miller-lab.github.io/AMSI2026_PGF/notebooks/BasicGenFxn/GenFxnsDiff.html)
8. [The Galton-Watson Branching Process, Aditya Singh, University of Chicago REU 2024](https://math.uchicago.edu/~may/REU2024/REUPapers/Singh.pdf)
9. [Bienayme Galton Watson Branching Process, Madhira & Deshmukh, Springer](https://ideas.repec.org/h/spr/sprchp/978-981-99-5601-2_5.html)
10. [The Basic M/G/1 Queueing System](https://ndl.ethernet.edu.et/bitstream/123456789/23430/1/273.pdf.pdf)
11. [A primer on the use of probability generating functions in infectious disease modeling, arXiv](https://arxiv.org/abs/1803.05136)
12. [Generating Functions, Statistics for Scientists and Engineers, Wiley](https://onlinelibrary.wiley.com/doi/10.1002/9781119047063.ch9)
13. [STAT 516 Lecture 1: Review on Probability and Statistics, University of Washington](https://faculty.washington.edu/yenchic/26A_stat516/Lec1_probability.html)
14. [MGFs vs Characteristic Functions, Explained in R](https://r-statistics.co/Moment-Generating-Functions-vs-Characteristic-Functions.html)
15. [Probability Generating Function, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/probability-generating-function)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability distributions › Characteristic and generating functions › Probability-generating functions*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license