Galton-Watson process
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The Galton-Watson process is a stochastic process arising from Francis Galton's statistical investigation of the extinction of surnames.
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History
There was concern amongst the Victorians that aristocratic surnames were becoming extinct. Galton originally posed the question regarding the probability of such an event in the Educational Times of 1837, and the Reverend Henry William Watson replied with a solution. Together, they then wrote an 1874 paper entitled On the probability of extinction of families. However, the concept was previously discussed by I. J. Bienaymé; see Heyde and Seneta 1977; though it appears that Galton and Watson derived their process independently. For a detailed history see Kendall (1966 and 1975)
Concepts
Assume, as was taken quite for granted in Galton's time, 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 surely die out, and if it is more than 1, then there is more than zero probability that it will survive forever.
Applications
Apart from the extinction of family problems, it also has other applications. For example, calculating the probability of extinction of a small population of organisms.
See also
References
- C C Heyde and E Seneta (1977) I.J. Bienayme: Statistical Theory Anticipated. Berlin, Germany.
- D G Kendall. (1966) Journal of the London Mathematical Society 41:385-406
- D G Kendall. (1975) Bulletin of the London Mathematical Society 7:225-253
External links
- On the Probability of the Extinction of Families (http://www.mugu.com/browse/galton/search/essays/pages/galton-1874-jaigi-family-extinction_1.htm)
- [1] (http://www-users.york.ac.uk/~pml1/stats/gwproc.ps) paper that took some of the above from.