Some distribution results on generalized ballot problems

Saran, Jagdish ; Sen, Kanwar

Applications of Mathematics, Tome 30 (1985) no. 3, pp. 157-165.

Résumé

Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $$\begin{gathered} \left(\begin{array}{c}a+b\end{array} \\ a\text{endmatrix}\right) \end{gathered}$$ voting sequences are equally probable. Denote by ${\alpha }_r$ and by ${\beta }_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1,\dots ,a+b$. The purpose of this paper is to derive, for $a\ge b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1,\dots ,a+b$ for which (i) ${\alpha }_r={\beta }_r-c$, (ii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$, (iii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c\pm 1$, where $c=0,\pm 1,\pm 2,\dots$.

Détail

DOI : 10.21136/AM.1985.104138

Classification : 60C05, 60E99, 60J15
Mots-clés : ballot problem