Editing Galton–Watson process (section)

{{Short description|Model for the extinction of family names}}
[[Image:Galton Watson survival Poisson.png|320px|thumb|Galton&ndash;Watson survival probabilities for different exponential rates of population growth, if the number of children of each parent node can be assumed to follow a [[Poisson distribution]].  For ''&lambda;''&nbsp;&le;&nbsp;1, eventual extinction will occur with probability&nbsp;1.  But the probability of survival of a new type may be quite low even if ''&lambda;''&nbsp;>&nbsp;1 and the population as a whole is experiencing quite strong [[exponential growth|exponential increase]].]]

The '''Galton–Watson process''', also called the '''Bienaymé-Galton-Watson process''' or the '''Galton-Watson branching process''', is a [[branching process|branching]] [[stochastic process]] arising from [[Francis Galton]]'s statistical investigation of the extinction of [[family name]]s.<ref>{{Cite book |last=Harris |first=Ted |author-link=Ted Harris (mathematician) |url=https://www.rand.org/content/dam/rand/pubs/reports/2009/R381.pdf |title=The theory of branching processes |date=1963 |publisher=Springer |isbn=978-3-642-51868-3 |edition=1st |location=Berlin |language=en}}</ref><ref>{{Citation |title=Galton–Watson branching processes |date=2009 |work=Epidemics and Rumours in Complex Networks |pages=7–18 |editor-last=Massoulié |editor-first=Laurent |url=https://www.cambridge.org/core/books/abs/epidemics-and-rumours-in-complex-networks/galtonwatson-branching-processes/A5131879FE6EFCC2BBA559988FE227A4 |access-date=2024-11-25 |series=London Mathematical Society Lecture Note Series |place=Cambridge |publisher=Cambridge University Press |doi=10.1017/CBO9780511806018.002 |isbn=978-0-521-73443-1 |editor2-last=Draief |editor2-first=Moez|url-access=subscription }}</ref> The process models family names as [[patrilineal]] (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants).

Galton's investigation of this process laid the groundwork for the study of [[branching processes]] as a subfield of [[probability theory]], and along with these subsequent processes the Galton-Watson process has found numerous applications across population genetics, computer science, and other fields.<ref>{{cite journal |last1=Jagers |first1=Peter |title=Branching processes as population dynamics |journal=Bernoulli |date=1995 |volume=1 |issue=1/2 |pages=191-200}}</ref>