Some distribution results on generalized ballot problems

Aplikace matematiky (1985)

• Volume: 30, Issue: 3, page 157-165
• ISSN: 0862-7940

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Abstract

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Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(a+ba\right)$ 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, $r=1,\cdots ,a+b$. 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 $r=1,\cdots ,a+b$ for which (i) ${\alpha }_{r}={\beta }_{r}-c$, (ii) ${\alpha }_{r}={\beta }_{r}-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c±1$, (iii) ${\alpha }_{r}={\beta }_{r}-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c±1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c±1$, where $c=0,±1,±2,\cdots$.

How to cite

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Saran, Jagdish, and Sen, Kanwar. "Some distribution results on generalized ballot problems." Aplikace matematiky 30.3 (1985): 157-165. <http://eudml.org/doc/15394>.

@article{Saran1985,
abstract = {Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(\{a+b\} \\ a \right)$ 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 $a\ge b-c$, 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 $c=0,\pm 1, \pm 2, \dots$.},
author = {Saran, Jagdish, Sen, Kanwar},
journal = {Aplikace matematiky},
keywords = {ballot problem},
language = {eng},
number = {3},
pages = {157-165},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some distribution results on generalized ballot problems},
url = {http://eudml.org/doc/15394},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Saran, Jagdish
AU - Sen, Kanwar
TI - Some distribution results on generalized ballot problems
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 3
SP - 157
EP - 165
AB - Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left({a+b} \\ a \right)$ 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 $a\ge b-c$, 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 $c=0,\pm 1, \pm 2, \dots$.
LA - eng
KW - ballot problem
UR - http://eudml.org/doc/15394
ER -

References

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