# For a p.g.f, of $X, f_X(s)$, is $s$ really “just” a dummy variable?

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.

• 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. – Kavi Rama Murthy Feb 1 at 0:02
• @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 – libby Feb 1 at 0:14
• 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. – Sangchul Lee Feb 1 at 0:32
• 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 – libby Feb 1 at 0:46