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The p.g.f, $f_X(s)$ of a random variable $X$ is given by

$$ f_X(s) = \sum_{x = 0}^\infty P(X = x)s^x$$

Usually, it's said the $s$ is just a dummy variable, but having been exposed to the result that the smallest root of $f(s) = s$ is the extinction probability of a branching process, it's clear that there is meaning to $s$, but I don't understand it's general meaning.

Any further intuition is highly appreciated.

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  • $\begingroup$ Whether you write the equation as $f(s)=s$ or $f(x)=x$ the solutions are the same. Your question has nothing to do with probability theory. It is a very basic question about the concept of a function. $\endgroup$ – Kavi Rama Murthy Feb 1 at 0:02
  • $\begingroup$ @KaviRamaMurthy please elaborate. $x$ is a fixed point of $f$, but what does this really mean in the context of the p.g.f, $f_X$? I don't understand $f_X$ in the context of a mapping $\endgroup$ – libby Feb 1 at 0:14
  • $\begingroup$ Without further context, a fixed point of a p.g.f. means nothing. For instance, it has certain interpretation in the context of branching process when $X$ provides the offspring distribution. $\endgroup$ – Sangchul Lee Feb 1 at 0:32
  • $\begingroup$ Right, so in this context, the p.g.f seems to have a meaningful interpretation because it's being used as an object to solve something meaningful about the population, which explains my question $\endgroup$ – libby Feb 1 at 0:46

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