Table 1.

A few common probability distributions and their PGFs.

Distribution	PGF $f\left(x\right)=\sum _i{r}_i{x}^i$
Poisson, mean λ: $r_i=\frac{e^{-\lambda }{\lambda }^i}{i!}$	$e^{\lambda \left(x-1\right)}$
Uniform: $r_{\lambda }=1$	$x^{\lambda }$
Binomial: n trials, with success probability p: $r_i=\left(\begin{array}{c}n\\ i\end{array}\right)p^i{q}^{n-i}$ for $q=1-p$	${[q+px]}^n$
Geometrica: $r_i=q^{i}p$ for $q=1-p$ and $i=\mathrm{0,1},\dots$	$p/\left(1-qx\right)$
Negative binomialb: $r_i=\left(\begin{array}{c}i+\hat{r}-1\\ i\end{array}\right)q^{\hat{r}}{p}^i$ for $q=1-p$	${\left(\frac{q}{1-px}\right)}^{\hat{r}}$

^aAnother definition of the geometric distribution with different indexing, $r_i=q^{i-1}p$ for $i=\mathrm{1,2},\dots$, gives a different PGF.

^bTypically the negative binomial is expressed in terms of a parameter r which is the number of failures at which the experiment stops, assuming each with success probability p. For us $r_i$ plays an important role, so to help distinguish these, we use $\hat{r}$ rather than r. Then $r_i$ is the probability of i successes.