Hostname: page-component-cb9f654ff-hqlzj Total loading time: 0 Render date: 2025-08-21T06:52:52.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Multimodality of the Markov Binomial Distribution

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Michel Dekking  and
Derong Kong
Michel Dekking*
Affiliation:
Delft University of Technology
Derong Kong*
Affiliation:
Delft University of Technology
*
∗ Postal address: Delft University of Technology, Faculty EWI, PO Box 5031, 2600 GA Delft, The Netherlands.
∗ Postal address: Delft University of Technology, Faculty EWI, PO Box 5031, 2600 GA Delft, The Netherlands.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closed-form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.

Keywords

Markov binomial distributionlog-concavitydouble peaking in kinetic transport

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 938 - 953
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Čekanavičius, V. and Roos, B. (2007). Binomial approximation to the Markov binomial distribution. Acta Appl. Math. 96, 137–146.Google Scholar
[2] Čekanavičius, V. and Roos, B. (2009). Poisson type approximations for the Markov binomial distribution. Stoch. Process. Appl. 119, 190–207.Google Scholar
[3] Daugman, J. (2003). The importance of being random: statistical principles of iris recognition. Pattern Recognition 36, 279–291.Google Scholar
[4] Dekking, M. and Hensbergen, A. (2009). A problem with the assessment of an iris identification system. SIAM Rev. 51, 417–422.Google Scholar
[5] Dekking, M. and Kong, D. (2011). A simple stochastic kinetic transport model. Preprint. Available at http://arxiv.orglabs/1106.2912v1.Google Scholar
[6] Dobrušin, R. L. (1953). Limit theorems for a Markov chain of two states. Izvestiya Akad. Nauk SSSR. Ser. Mat. 17, 291–330 (in Russian). English translation: Select. Transl. Math. Statist. Prob. 1 (1961), pp. 97-134.Google Scholar
[7] Edwards, A. W. F. (1960). The meaning of binomial distribution. Nature 186, 1074.Google Scholar
[8] Gabriel, K. R. (1959). The distribution of the number of successes in a sequence of dependent trials. Biometrika 46, 454–460.Google Scholar
[9] Helgert, H. J. (1970). On sums of random variables defined on a two-state Markov chain. J. Appl. Prob. 7, 761–765.Google Scholar
[10] Liu, L. L. and Wang, Y. (2007). A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38, 542–560.Google Scholar
[11] Markov, A. A. (1924). Probability Theory, 4th edn. Moscow (in Russian).Google Scholar
[12] Michalak, A. A. M. and Kitanidis, P. K. (2000). Macroscopic behavior and random-walk particle tracking of kinetically sorbing solutes. Water Resources Res. 36, 2133–2146.Google Scholar
[13] Omey, E., Santos, J. and Van Gulck, S. (2008). A Markov-binomial distribution. Appl. Anal. Discrete Math. 2, 38–50.Google Scholar
[14] Pitman, J. (1997). Probabilistic bounds on the coefficients of polynomials with only real zeros. J. Combinatorial Theory A 77, 279–303.Google Scholar
[15] Viveros, R., Balasubramanian, K. and Balakrishnan, N. (1994). Binomial and negative binomial analogues under correlated Bernoulli trials. Amer. Statistician 48, 243–247.Google Scholar
[16] Wang, Y. H. (1981). On the limit of the Markov binomial distribution. J. Appl. Prob. 18, 937–942.Google Scholar
[17] Wang, Y. and Yeh, Y.-N. (2007). Log-concavity and LC-positivity. J. Combinatorial Theory A 114, 195–210.Google Scholar