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What does it mean when I plug in $0, 1, ..., n$ into a probability generating function?

What does a probability generating function tell me about the number I plugged in?

kjetil b halvorsen
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Rahil
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1 Answers1

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Probability generating functions have multiple desirable properties; however, they only work for discrete distributions, which belong to random variables that can only assume a finite or countably infinite number of values - e.g. $\{0,1,\dots,n\}.$ This explains the first part of your question.

Instead of going over all the properties of PGF's, let me just point out the feature that is easiest to appreciate:

Differentiating the PDF, and evaluating at zero gives you actual probabilities:

If we annotate the PGF as

$$ G(z)=\operatorname{E} (z^{X})=\sum_{x=0}^{\infty }p(x)z^{x},$$

with $p(x)$ corresponding to the probability mass function of $X$, you can find the probability of a particular outcome $X=k$ as:

$$ \Pr(X=k) = \frac{1}{k!}\frac{d^k \, G(z)}{dx^k} \vert_{z=0}=\frac{1}{k!}G^{(i)}(0).$$

Antoni Parellada
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    This property relies on the assumption that the variable's support is a subset of the non-negative integers. For a concrete example of where your formula can otherwise be incorrect, please visit http://stats.stackexchange.com/questions/103969. – whuber Mar 06 '17 at 21:53
  • You say `however, they only work for discrete distributions` which is not completely true, see https://stats.stackexchange.com/questions/297711/what-is-the-difference-between-moment-generating-function-and-probability-genera/297713#297713 – kjetil b halvorsen Nov 03 '19 at 19:32