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Exact and limiting distributions of the number of lead positions in “unconditional” ballot problems

Published online by Cambridge University Press:  14 July 2016

Ora Engelberg
Ora Engelberg*
Affiliation:
Columbia University, New York.
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Extract

In a ballot, candidate A scores a votes and candidate B scores b votes. Suppose the ballots are drawn out one at a time, and denote αr and βr the number of votes registered for A and B, respectively, among the first r votes recorded. Further, let Δa,b be the number of subscripts r satisfying the strict lead condition , let be the number of subscripts r satisfying the weak lead condition ; and suppose all possible () voting records are equally probable. The probability distributions of the number of strict and weak lead positions corresponding to and , respectively, have been determined in [4] for a≧b.

Type
Short Communications
Information
Journal of Applied Probability , Volume 1 , Issue 1 , June 1964 , pp. 168 - 172
Copyright
Copyright © Applied Probability Trust 1964 

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References

[1] Andersen, E. Sparre (1950) On the frequency of positive partial sums of a series of random variables. Mat. Tidsskr. B, 33–35.Google Scholar
[2] Andersen, E. Sparre (1953) On sums of symmetrically dependent random variables. Skand. Aktuartidskr. 36, 123–138.Google Scholar
[3] Chung, K. L. and Feller, W. (1949) Fluctuation in coin tossing. Proc. Nat. Acad. Sci. U.S.A. 35, 605–608.Google Scholar
[4] Engelberg, O. (1963) Generalizations of the ballot problem. Submitted to Zeit. Wahrscheinlichkeitstheorie. Google Scholar
[5] Erdös, P. and Kac, M. (1947) On the number of positive sums of independent random variables. Bull. Amer. Math. Soc. 53, 1011–1020.Google Scholar