Editing Galton–Watson process (section)

== Concepts ==

Assume, for the sake of the model, that surnames are passed on to all male children by their father.  Suppose the number of a man's sons to be a [[random variable]] [[probability distribution|distributed]] on the set&nbsp;{&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...&nbsp;}.  Further suppose the numbers of different men's sons to be [[statistical independence|independent]] random variables, all having the same distribution.

Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will [[almost surely]] die out, and if it is more than&nbsp;1, then there is more than zero probability that it will survive for any given number of generations.

A corollary of high extinction probabilities is that if a lineage ''has'' survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.{{Citation needed|date=November 2024}}