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My understanding of PGF is that it is just an efficient way to find the properties of the distribution (mean variance etc) and represents the whole distribution, and there isn’t really any meaning to the function itself. But then I saw Galton Watson Process (or branching process) and that G(q)=q for smallest non negative q is the extinction probability. I can’t help but think, is there actually some meaning to the PGF to make this true?

question Is there any physical meaning of the probability generating function like the mean or variance. (Context above)

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The correct statement, viz. Proposition 3 here, is that the least $q\ge0$ with $G(q)=q$ is the extinction probability.

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