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The number of individuals in a cascade process

Published online by Cambridge University Press:  24 October 2008

I. J. Good
I. J. Good
Affiliation:
131 Cheviot GardensLondon, N.W. 2
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Extract

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let

so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 3 , July 1949 , pp. 360 - 363
Copyright
Copyright © Cambridge Philosophical Society 1949

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References

REFERENCES

(1)Fisher, R. A.The Genetical Theory of Natural Selection (1930), p. 74 and Proc. Roy. Soc. Edinburgh, 42 (1922), 321–41.Google Scholar
(2)Galton, F.Natural Inheritance (1889), Appendix F by the Rev. H. W. Watson, F.R.S.Google Scholar
(3)Woodward, P. M.A statistical theory of cascade multiplication. Proc. Cambridge Phil. Soc. 44 (1948), 404–12.CrossRefGoogle Scholar