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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?

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    $\begingroup$ This is more commonly called "probability generating function" $\endgroup$ – kjetil b halvorsen Mar 14 '16 at 21:12
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This is called the probability generating function. The relationship between the pgf and mgf is:

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

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