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The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. 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). This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor. History There was concern amongst the Victorians that aristocratic surnames were becoming extinct. 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:A large nation, of whom we will only concern ourselves with adult males, N in number, and who each bear separate surnames colonise a district. Their law of population is such that, in each generation, a0 per cent of the adult males have no male children who reach adult life; a1 have one such male child; a2 have two; and so on up to a5 who have five. Find what proportion of their surnames will have become extinct after r generations; and how many instances there will be of the surname being held by m persons.The Reverend Henry William Watson replied with a solution. 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). Galton and Watson appear to have derived their process independently of the earlier work by I. J. Bienaymé; see. Their solution is incomplete, according to which all family names go extinct with probability 1. Bienaymé published the answer to the problem in 1845, with a promise to publish the derivation later, however there is no known publication of his solution. (However, Bru (1991) purports to reconstruct the proof). He was inspired by Émile Littré and Louis-François Benoiston de Châteauneuf (a friend of Bienaymé). Cournot published a solution in 1847, in chapter 36 of De l'origine et des limites de la correspondance entre l'algèbre et la géométrie. The problem in his formulation is Consider a gambler who buys lotteries. Each lottery costs 1 ecu and pays 0 , 1 , . . . , n {\displaystyle 0,1,...,n} with probabilities p 0 , . . . , p n {\displaystyle p_{0},...,p_{n}} . The gambler always spends all their money to buy lotteries. If the gambler starts with k {\displaystyle k} dollars, what's the probability of going bankrupt? 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. Haldane solved the problem in 1927. 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. Steffensen solved it in 1930. For a detailed history, see Kendall (1966 and 1975) and and also Section 17 of. 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 distributed on the set { 0, 1, 2, 3, ... }. Further suppose the numbers of different men's sons to be 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 1, then there is more than zero probability that it will survive for any given number of generations. Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or the chances of extinction of small population of organisms; as well as explaining (perhaps closest to Galton's original interest) why only a handful of males in the deep past of humanity now have any surviving male-line descendants, reflected in a rather small number of distinctive human Y-chromosome DNA haplogroups. 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. Mathematical definition A Galton–Watson process is a stochastic process {Xn} which evolves according to the recurrence formula X0 = 1 and X n + 1 = ∑ j = 1 X n ξ j ( n ) {\displaystyle X_{n+1}=\sum _{j=1}^{X_{n}}\xi _{j}^{(n)}} where { ξ j ( n ) : n , j ∈ N } {\displaystyle \{\xi _{j}^{(n)}:n,j\in \mathbb {N} \}} is a set of independent and identically-distributed natural number-valued random variables. In the analogy with family names, Xn can be thought of as the number of descendants (along the male line) in the nth generation, and ξ j ( n ) {\displaystyle \xi _{j}^{(n)}} can be thought of as the number of (male) children of the jth of these descendants. The recurrence relation states that the number of descendants in the n+1s.... Discover the Watson G Branch popular books. Find the top 100 most popular Watson G Branch books.