×

On distribution of the leadership time in counting votes and predicting winners. (English) Zbl 1397.60024

Summary: In two-candidate election the votes are counted in random order. Suppose that candidate \(A\) was the leader until the 9th vote. How may we use this information in predicting the future winner? To this aim we derive distributions of the first leadership time both for the winner and loser. Our conclusion is rather surprising.

MSC:

60C05 Combinatorial probability
62E15 Exact distribution theory in statistics
05A19 Combinatorial identities, bijective combinatorics
Full Text: DOI

References:

[1] Balakrishnan, N.; Koutras, M. V., Runs and Scans with Applications (2001), Wiley: Wiley New York · Zbl 0991.62087
[2] Bolfarine, H.; Zacks, S., Bayes and minimax prediction in finite populations, J. Statist. Plann. Inference, 28, 139-151 (1991) · Zbl 0728.62012
[3] Bolfarine, H.; Zacks, S., Prediction Theory for Finite Populations (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0751.62003
[4] Brémaud, P., An Introduction to Probabilistic Modeling (1994), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, Corrected 2nd printing
[5] Feller, W., An Introduction to Probability Theory and its Applications, Vol. 1 (1968), Wiley: Wiley New York · Zbl 0155.23101
[6] Ghosh, M.; Meeden, G., Bayesian Methods for Finite Population Sampling (1997), Chapman & Hall: Chapman & Hall London · Zbl 0894.62012
[7] Goulden, I. P.; Serrano, L. G., Maintaining the spirit of the reflection principle when the boundary has arbitrary integer slope, J. Combin. Theory Ser. A, 104, 317-326 (2003) · Zbl 1032.05004
[8] Guo, V. J.W., On Jensen’s and related combinatorial identities, Appl. Anal. Discrete Math., 5, 201-211 (2011) · Zbl 1265.05027
[9] Khan, R. A., A note on the generating function of a negative hypergeometric distribution, Sankhyā, 56, 309-313 (1994) · Zbl 0842.60015
[10] Lengyel, T., Direct consequences of the basic Ballot Theorem, Statist. Probab. Lett., 81, 1476-1481 (2011) · Zbl 1228.60055
[11] Meeden, G., A decision theoretic approach to imputation in finite population sampling, J. Amer. Statist. Assoc., 95, 586-595 (2000) · Zbl 0995.62007
[12] Miller, G. K.; Fridell, S. L., A forgotten discrete distribution? Reviving the negative hypergeometric model, Amer. Statist., 61, 347-350 (2007)
[13] Renault, M., Four proofs of the Ballot Theorem, Math. Mag., 80, 345-352 (2007) · Zbl 1144.05303
[14] Sen, K.; Agarwall, M.; Bhattacharya, G. N., Pólya-Eggenber F-S models of order \((k_1, k_2)\), Studia Sci. Math. Hungar., 43, 1-31 (2006) · Zbl 1174.60302
[15] Stępniak, C., On distribution of waiting time for the first failure followed by a limited length success run, Appl. Math. (Warsaw), 40, 421-430 (2013) · Zbl 1281.62237
[16] Takacs, L., On the ballot theorem, (Advances in Combinatorial Methods and Applications to Probability and Statistics (1997), Birkhauser), 97-114 · Zbl 0888.60012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.