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For a Galton-Watson branching process, if $f$ is the p.g.f of the process, then the extinction probability, $q$, is given by the root to $f(q) = q$.

This can be seen intuitively simply by writing out, if $X$ is the progeny of the root ancestor:

$$q = \sum_{i = 0}^{\infty} q^i P(X = i)$$

This gives the probability that each of the $i$ descendants goes extinct scaled by the probability of having $i$ descendants.

Is there a deeper intuition to the p.g.f for understanding why the extinction probability is given so elegantly? Is there more going on with $f(s), s \in [0,1]$?

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