Geodesic
Some distribution results on generalized ballot problems
Applications of Mathematics, Tome 30 (1985) no. 3, pp. 157-165
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Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(\matrix {a+b} \\ a \endmatrix \right)$ 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 $a\geq b-c$, 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 $c=0,\pm 1, \pm 2, \dots$.
Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(\matrix {a+b} \\ a \endmatrix \right)$ 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 $a\geq b-c$, 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 $c=0,\pm 1, \pm 2, \dots$.
DOI : 10.21136/AM.1985.104138
Classification : 60C05, 60E99, 60J15
Keywords: ballot problem 
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Saran, Jagdish; Sen, Kanwar. Some distribution results on generalized ballot problems. Applications of Mathematics, Tome 30 (1985) no. 3, pp. 157-165. doi: 10.21136/AM.1985.104138

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