DISTRIBUTIONS OF BALLOT PROBLEM RANDOM-VARIABLES

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes, and that all the possible voting records are equally probable. Corresponding to the first r votes, let alpha-r and beta-r be the numbers of votes registered for A and B, respectively. Let rho be an arbitrary positive real number. Denote by delta (a, b, rho)[delta*(a, b, rho)] the number of values of r fo which the inequality alpha-r greater-than-or-equal-to rho-beta-r[alpha-r > rho-beta-r], r = 1, ..., a + b, holds. Heretofore the probability distributions of delta and delta-* have been derived for only a restricted set of values of a, b, and rho, although, as pointed out here, they are obtainable for all values of (a, b, and rho) by using a result of Takacs (1964). In this paper we present a derivation of the distribution of delta [delta*] whose development, for any (a, b, rho), leads to both necessary and sufficient conditions for delta [delta*] to have a discrete uniform distribution.