I have a reference that gives an example of a Poisson "product moment generating function":

$$G(t)=E(t^X)=\sum_{i=0}^{\infty}t^i\frac{\theta^i}{n!}e^{-\theta}=\sum_{i=0}^{\infty}\frac{(\theta t)^i}{n!}e^{-\theta}=e^{\theta t}e^{-\theta}=e^{-\theta(1-t)}$$

Is this a common concept, since I only find "moment generating function" and how's this different from moment generating function? Specifically, why is there the $n!$ and not $i!$ (as per the p.d.f of Poisson) in the denominator?

  • 2
    $\begingroup$ This is more commonly called "probability generating function" $\endgroup$ Commented Mar 14, 2016 at 21:12

1 Answer 1


This is called the probability generating function. The relationship between the pgf and mgf is:

$G_X(e^t) = M_X(t)$, where

  • $G_X(t) := E[t^X]$

  • $M_X(t) := E[e^{tX}]$


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