Previous |  Up |  Next

Article

Keywords:
ballot problem
Summary:
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 $c=0,\pm 1, \pm 2, \dots$.
References:
[1] A. Aeppli: Zur Theorie Verketteter Wahrscheinlichkeiten. Thèse, Zürich (1924).
[2] D. André: Solution directe du problème rèsolu par M. Bertrand. C. R. Acad. Sci. (Paris), 105 (1887), 436-437.
[3] É. Barbier: Généralisation du problème rèsolu par M. J. Bertrand. C. R. Acad. Sci. (Paris), 105 (1887), 407.
[4] J. Bertrand: Solution ďun probléme. C. R. Acad. Sci. (Paris), 105 (1887), 369.
[5] M. T. L. Bizley: Derivation of a new formula for the number of minimal lattice paths from $(0, 0)$ to $(km, kn)$ having just t contacts with the line $my = nx$ and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuar., 80 (1954), 55-62. MR 0061567
[6] M. T. L. Bizley: Problem 5503. Amer. Math. Monthly, 74 (1967), 728.
[7] K. L. Chung W. Feller: Fluctuations in coin tossing. Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 605-608. DOI 10.1073/pnas.35.10.605 | MR 0033459
[8] A. Dvoretzky, Th. Motzkin: A problem of arrangements. Duke Math. Journal, 14 (1947), 305-313. DOI 10.1215/S0012-7094-47-01423-3 | MR 0021531
[9] O. Engelberg: Exact and limiting distributions of the number of lead positions in 'unconditional' ballot problems. J. Appl. Prob., 1 (1964), 168-172. DOI 10.2307/3212068 | MR 0161354 | Zbl 0203.19301
[10] O. Engelberg: Generalizations of the ballot problem. Z. Wahrscheinlichkeitstheorie, 3 (1965), 271-275. DOI 10.1007/BF00535777 | MR 0185626 | Zbl 0131.17304
[11] W. Feller: An introduction to probability theory and its Applications. Vol. I., Third Edition, John Wiley, New York (1968). MR 0228020 | Zbl 0155.23101
[12] H. D. Grossman: Another extension of the ballot problem. Scripta Math., 16 (1950), 120-124.
[13] S. G. Mohanty T. V. Narayana: Some properties of compositions and their application to probability and statistics I. Biometrische Zeitschrift, 3 (1961), 252-258. DOI 10.1002/bimj.19610030403
[14] L. Takács: A generalization of the ballot problem and its application in the theory of queues. J. Amer. Statist. Assoc., 57 (1962), 327-337. MR 0138139
[15] L. Takács: Ballot problems. Z. Wahrscheinlichkeitstheorie, 1 (1962), 154-158. DOI 10.1007/BF01844418 | MR 0145601
[16] L. Takács: The distribution of majority times in a ballot. Z. Wahrscheinlichkeitstheorie, 2 (1963), 118-121. DOI 10.1007/BF00531965 | MR 0160276
[17] L. Takács: Fluctuations in the ratio of scores in counting a ballot. J. Appl. Prob., 1 (1964), 393-396. DOI 10.2307/3211869 | MR 0169340
[18] L. Takács: Combinatorial methods in the theory of stochastic processes. John Wiley, New York (1967). MR 0217858
[19] L. Takács: On the fluctuations of election returns. J. Appl., Prob., 7 (1970), 114-123. DOI 10.2307/3212153 | MR 0253447
Partner of
EuDML logo