Distributions of ballot problem random variables

Abstract

Suppose that in a ballot candidateAscoresavotes and candidateBscoresbvotes, and that all the possible voting records are equally probable. Corresponding to the firstrvotes, letαrandβrbe the numbers of votes registered forAandB, respectively. Let p be an arbitrary positive real number. Denote byδ(a, b, p)[δ*(a,b,ρ)] the number of values ofrfor which the inequality,r =1, ···,a+b, holds. Heretofore the probability distributions of δand δ* have been derived for only a restricted set of values ofa, b, andρ, although, as pointed out here, they are obtainable for all values of (a,b,ρ) by using a result of Takács (1964). In this paper we present a derivation of the distribution ofδ[δ*] whose development, for any (a, b, ρ), leads to both necessary and sufficient conditions forδ[δ*] to have a discrete uniform distribution.