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Sums of random variables having the modified geometric distribution by application to two-person games

Published online by Cambridge University Press:  01 July 2016

M. J. Phillips
M. J. Phillips*
Affiliation:
University of Leicester
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Abstract

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.

Keywords

MODIFIED GEOMETRIC DISTRIBUTIONCOMPOUND DISTRIBUTIONNEGATIVE BINOMIAL DISTRIBUTIONPASCAL DISTRIBUTIONTWO-PERSON GAME
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 3 , September 1978 , pp. 647 - 665
Copyright
Copyright © Applied Probability Trust 1978 

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References

Anderson, C. L. (1977) Note on the advantage of first serve. J. Combinatorial Theory A 23, 363.Google Scholar
Carter, W. H. and Crews, S. L. (1974) An analysis of the game of tennis. Amer. Statistician 28, 130–134.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I. Wiley, New York.Google Scholar
George, S. L. (1973) Optimal strategy in tennis: a simple probabilistic model. Appl. Statist. 22, 97–103.CrossRefGoogle Scholar
Goad, R. (1973) How fair is British scoring? The Squash Player 3, 24–26. HsiB. P. and BurychD. M. (1971) Games of two players. Appl. Statist. 20, 86–92.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Discrete Distributions. Houghton Mifflin, Boston, Mass. Google Scholar
Kingston, J. G. (1976) Comparison of scoring systems in two-sided competitions. J. Combinatorial Theory A 20, 357–362.Google Scholar
Patil, G. G. and Joshi, S. W. (1968) A Dictionary and Bibliography of Discrete Distributions. Oliver and Boyd, Edinburgh.Google Scholar
Official Rules of Sports and Games, 1976–1977 (1976) Kaye and Ward, London.Google Scholar