Editing Galton–Watson process (section)

==History==
There was concern amongst the [[Victorian era|Victorian]]s that [[Aristocracy (class)|aristocratic]] surnames were becoming extinct.<ref>{{cite book |last1=Lysons |first1=Daniel |last2=Lysons |first2=Samuel |title=General history: Extinct noble families |chapter=In Magna Britannia: Volume 6, Devonshire |year=1822 |publisher=British History Online |url=https://www.british-history.ac.uk/magna-britannia/vol6/xcv-cviii |access-date=25 November 2024}}</ref>

In 1869, Galton published ''[[Hereditary Genius]]'', in which he treated the extinction of different social groups.

Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of ''[[The Educational Times]]:''<ref>{{cite journal |author=[[Francis Galton]] |title=Problem 4001 |journal=[[Educational Times]] |volume=25 |issue=143 |date=1873-03-01 |url=http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF |url-status=dead |archiveurl=https://web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF |archivedate=2017-01-23 |page=300}}</ref><blockquote>A large nation, of whom we will only concern ourselves with the adult males, {{math|N}} in number, and who each bear separate surnames colonise a district. Their law of population is such that, in each generation, {{math|''a''{{sub|0}}}} per cent. of the adult males have no male children who reach adult life ; {{math|''a''{{sub|1}}}} have one such male child ; {{math|''a''{{sub|2}}}} have two ; and so on up to {{math|''a''{{sub|5}}}} who have five. Find (1) what proportion of their surnames will have become extinct after {{mvar|r}} generations ; and (2) how many instances there will be of the surname being held by {{mvar|m}} persons.</blockquote>The Reverend [[Henry William Watson]] replied with a solution.<ref>{{cite journal |author=[[Henry William Watson]] |date=1873-08-01 |title=Problem 4001 |url=http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF |url-status=dead |journal=[[Educational Times]] |volume=26 |issue=148 |page=115 |archiveurl=https://web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF |archivedate=2016-12-01}}<br /> A first offering submitted by G.S. Carr, according to Galton, was "totally erroneous"; see {{cite journal |author=G. S. Carr |date=1873-04-01 |title=Problem 4001 |url=http://ioearc.da.ulcc.ac.uk/9305/1/Educational%20Times%20Vol%2026%20Iss%20144.PDF |url-status=dead |journal=[[Educational Times]] |volume=26 |issue=144 |page=17 |archiveurl=https://web.archive.org/web/20170803094523/http://ioearc.da.ulcc.ac.uk/9305/1/Educational%20Times%20Vol%2026%20Iss%20144.PDF |archivedate=2017-08-03}}</ref> Together, they then wrote an 1874 paper titled "On the probability of the extinction of families" in the ''Journal of the Anthropological Institute of Great Britain and Ireland'' (now the ''[[Journal of the Royal Anthropological Institute]]'').<ref>Galton, F., & Watson, H. W. (1875). [https://www.jstor.org/stable/pdf/2841222.pdf "On the probability of the extinction of families"]. ''[[Journal of the Royal Anthropological Institute]]'', '''4''', [https://babel.hathitrust.org/cgi/pt?id=mdp.39015027596207;view=1up;seq=166 138–144].</ref> Galton and Watson appear to have derived their process independently of the earlier work by [[Irénée-Jules Bienaymé|I. J. Bienaymé]]; see.<ref>{{Cite journal |author1-link=Chris Heyde |last1=Heyde |first1=C. C.|author2-link=Eugene Seneta |last2=Seneta |first2=E. |date=1972 |title=Studies in the History of Probability and Statistics. XXXI. The simple branching process, a turning point test and a fundamental inequality: A historical note on I. J. Bienaymé |url=https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/59.3.680 |journal=Biometrika |language=en |volume=59 |issue=3 |pages=680–683 |doi=10.1093/biomet/59.3.680 |issn=0006-3444|url-access=subscription }}</ref> Their solution is incomplete, according to which ''all'' family names go extinct with probability 1.

[[Irénée-Jules Bienaymé|Bienaymé]] had previously published the answer to the problem in 1845,<ref>Bienayme, I.J. (1845). ''[https://probabilityandfinance.com/pulskamp/Bienayme/trans%201845.pdf De la loi de multiplication et de la durée des families]''. L’Institut, 589, Vol. 13, pp. 131–132. Soc. Philomat. Paris Extraits, Ser. 5, 37–39. (Reprinted in Kendall, D. G. (1975))</ref> with a promise to publish the derivation later, however there is no known publication of his solution. (However, Bru (1991)<ref>Bru, Bernard. "A la recherche de la démonstration perdue de Bienaymé." ''Mathématiques et Sciences humaines'' 114 (1991): 5-17.</ref> purports to reconstruct the proof). He was inspired by [[Émile Littré]]<ref>Littré, Émile. ''Analyse raisonnée du cours de philosophie positive de M. Auguste Comte''. 1845.</ref> and [[:fr:Louis-François Benoiston de Châteauneuf|Louis-François Benoiston de Châteauneuf]] (a friend of Bienaymé).<ref>L. F. Benoiston de Châteauneuf, "''Sur la durée des familles nobles de France''," Séances et Travaux de l'Académie des Sciences Morales et Politiques: Comptes Rendus, 7 (1845), 210-240.</ref><ref name=":1" />

[[Antoine Augustin Cournot|Cournot]] published a solution in 1847, in Chapter V, §36 of ''De l'origine et des limites de la correspondance entre l'algèbre et la géométrie''.<ref>{{Cite book |last=Cournot |first=A. A. (Antoine Augustin) |url=http://archive.org/details/delorigineetdesl00cour |title=De l'origine et des limites de la correspondance entre l'algèbre et la géométrie |date=1847 |publisher=Paris : L. Hachette |others=University of Illinois Urbana-Champaign}}</ref> The problem in his formulation is the following: consider a gambler who buys lotteries. Each lottery costs 1 [[écu]] and pays {{math|0, 1, ..., ''m''}} écus with probabilities {{math|''k''{{sub|0}}, ''k''{{sub|1}}, ..., ''k{{sub|m}}''}}, respectively. The gambler starts with 1 écu before round 1, and at each round of gambling spends all their money to buy lotteries. Let {{math|''p{{sub|n}}''}} denote the probability that the gambler goes backrupt before the {{math|(''n''+1)}}-th round of gambling. What is the limit of {{math|''p{{sub|n}}''}}?

Ronald A. Fisher in 1922 studied the same problem formulated in terms of genetics. Instead of the extinction of family names, he studied the probability for a mutant gene to eventually disappear in a large population.<ref>{{Cite journal |last=Fisher |first=R. A. |date=1922 |title=XXI.—On the Dominance Ratio |url=https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh/article/abs/xxion-the-dominance-ratio/2CDBE1D374EBE9B2774DF301BA98A584 |journal=Proceedings of the Royal Society of Edinburgh |language=en |volume=42 |pages=321–341 |doi=10.1017/S0370164600023993 |issn=0370-1646|url-access=subscription }}</ref> [[J. B. S. Haldane|Haldane]] solved the problem in 1927.<ref>{{Cite journal |last=Haldane |first=J. B. S. |authorlink=J. B. S. Haldane |date=July 1927 |title=A Mathematical Theory of Natural and Artificial Selection, Part V: Selection and Mutation |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/mathematical-theory-of-natural-and-artificial-selection-part-v-selection-and-mutation/9B6F4FE68136A70E06133E2E389EFA5B |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |language=en |volume=23 |issue=7 |pages=838–844 |doi=10.1017/S0305004100015644 |bibcode=1927PCPS...23..838H |s2cid=86716613 |issn=1469-8064|url-access=subscription }}</ref>

[[Agner Krarup Erlang]] was a member of the prominent Krarup family, which was going extinct. In 1929, he published the same problem posthumously (his obituary appears beside the problem). Erlang died childless. [[Johan Frederik Steffensen|Steffensen]] solved it in 1930.

For a detailed history, see Kendall (1966<ref>{{Cite journal |last=Kendall |first=David G. |date=1966 |title=Branching Processes Since 1873 |url=http://doi.wiley.com/10.1112/jlms/s1-41.1.385 |journal=Journal of the London Mathematical Society |language=en |volume=s1-41 |issue=1 |pages=385–406 |doi=10.1112/jlms/s1-41.1.385|url-access=subscription }}</ref> and 1975<ref name=":1">{{Cite journal |last=Kendall |first=David G. |date=November 1975 |title=The Genealogy of Genealogy Branching Processes before (and after) 1873 |url=http://doi.wiley.com/10.1112/blms/7.3.225 |journal=Bulletin of the London Mathematical Society |language=en |volume=7 |issue=3 |pages=225–253 |doi=10.1112/blms/7.3.225|url-access=subscription }}</ref>) and<ref>{{Cite journal |last=Albertsen |first=K. |date=1995 |title=The Extinction of Families |url=https://www.jstor.org/stable/1403617 |journal=International Statistical Review / Revue Internationale de Statistique |volume=63 |issue=2 |pages=234–239 |doi=10.2307/1403617 |jstor=1403617 |s2cid=124630211 |issn=0306-7734|url-access=subscription }}</ref> and also Section 17 of.<ref name=":0">{{Cite journal |last1=Simkin |first1=M. V. |last2=Roychowdhury |first2=V. P. |date=2011-05-01 |title=Re-inventing Willis |url=https://www.sciencedirect.com/science/article/pii/S0370157310003339 |journal=Physics Reports |language=en |volume=502 |issue=1 |pages=1–35 |doi=10.1016/j.physrep.2010.12.004 |arxiv=physics/0601192 |bibcode=2011PhR...502....1S |s2cid=88517297 |issn=0370-1573}}</ref>