Informatics and Applications
2016, Volume 10, Issue 4, pp 1120
THE POISSON THEOREM FOR BERNOULLI TRIALS WITH A RANDOM PROBABILITY OF SUCCESS AND A DISCRETE ANALOG OF THE WEIBULL DISTRIBUTION
 V. Yu. Korolev
 A. Yu. Korchagin
 A. I. Zeifman
Abstract
A problem related to the Bernoulli trials with a random probability of success is considered. First, as a result of the preliminary experiment, the value of the random variable ï ˆ (0,1) is determined that is taken as the probability of success in the Bernoulli trials. Then, the random variable N is determined as the number of successes in k ˆ N Bernoulli trials with the so determined success probability ï. The distribution of the random variable N is called ïmixed binomial. Within the framework of these Bernoulli trials with the random probability of success, a "random" analog of the classical Poisson theorem is formulated for the ïmixed binomial distributions, in which the limit distribution turns out to be the mixed Poisson distribution. Special attention is paid to the case where mixing is performed with respect to the Weibull distribution. The corresponding mixed Poisson distribution called PoissonWeibull law is proposed as a discrete analog of the Weibull distribution. Some properties of the PoissonWeibull distribution are discussed. In particular, it is shown that this distribution can be represented as the mixed geometric distribution. A twostage grid algorithm is proposed for estimation of parameters of mixed Poisson distributions and, in particular, of the PoissonWeibull distribution. Statistical estimators for the upper bound of the grid are constructed. The examples of practical computations performed by the proposed algorithm are presented.
[+] References (19)
 Grigoryeva, M. E., V. Yu. Korolev, and I. A. Sokolov.
2013. Predel'naya teorema dlya geometricheskikh summ nezavisimykh neodinakovo raspredelennykh sluchaynykh velichin i ee primenenie k progrozirovaniyu veroyatnos ti katastrof v neodnorodnykh potokakh ekstremal'nykh sobytiy [A limit theorem for geometric sums of independent nonidentically distributed random variables and its application to the prediction of the probabilities of catastrophes in nonhomogeneous flows of extremal events]. Informatika i ee Primeneniya  Inform. Appl. 7(4):1119.
 Renyi, A. 1956. A Poissonfolyamat egy jellemzese. Maguar Tud. Acad. Mat. Int. Kozl. 1:519527.
 Mogyorodi, J. 1971. Some notes on thinning recurrent flows. Litovsky Math. Sbornik 11:303315.
 Zolotarev, V. M. 1983. Odnomernye ustoychivye rasprede leniya [Onedimensional stable distributions]. Moscow: Nauka. 304 p.
 Schneider, W. R. 1986. Stable distributions: Fox function representationand generalization. Stochastic processes in classical and quantum systems. Eds. S. Albeverio, G. Casati, and D. Merlini. Berlin: Springer. 497511.
 Uchaikin, V. V., and V. M. Zolotarev. 1999. Chance and stability. Utrecht: VSP. 570 p.
 Korolev, V. Yu., V. E. Bening, and S.Ya. Shorgin. 2011. Matematicheskie osnovy teorii riska [Mathematical fundamentals of risk theory]. 2nd ed. Moscow: Fizmatlit. 591 p.
 Grandell, J. 1997. Mixed Poisson processes. London: Chapman and Hall. 268 p.
 Korolev, V. Yu., A. Yu. Korchagin, and A. I. Zeifman. 2017 (in press). On doubly stochastic rarefaction of renewal processes. 14th Conference (International) of Numerical Analysis and Applied Mathematics Proceedings. American Institute of Physics Proceedings.
 Nakagawa, T, and Sh. Osaki. 1975. The discrete Weibull distribution. IEEE Trans. Reliab. 24:300301.
 Laherrere, J., and D. Sornette. 1998. Stretched exponential distributions in nature and economy: "Fat tails" with characteristic scales. Eur. Phys. J. B 2:525539.
 Malevergne, Y., V. Pisarenko, and D. Sornette. 2005. Empirical distributions of stock returns: Between the stretched exponential and thepower law? Quant. Financ. 5:379401.
 Malevergne, Y., V. Pisarenko, and D. Sornette. 2006. On the power of generalized extreme value (GEV) and generalized Pareto distribution (GDP) estimators for empirical distributions of stock returns. Appl. Financ. Econ. 16:271289.
 Korolev, V. Yu. 2016. Product representations for random variables with the Weibull distributions and their applications. J. Math. Sci. 218(3):298313.
 Karlis, D. 2005. An EM algorithm for mixed Poisson distributions. ASTINBull. 35:324.
 Korolev, V. Yu., and A. Yu. Korchagin. 2014. Modifi tsirovannyy setochnyy metod razdeleniya dispersionno sdvigovykh smesey normal'nykh zakonov [Modified grid method for decomposition of meanvariance normal mixtures]. Informatika i ee Primeneniya Inform. Appl. 8(4):1119.
 Korolev, V. Yu. 2011. Veroyatnostnostatisticheskie metody dekompozitsii volatil'nosti khaoticheskikh protsessov [Probablitybased method for volatility decomposition of chaotic processes]. Moscow: Moscow University Press. 510 p.
 Korolev, V. Yu., and A. L. Nazarov. 2010. Separating mixtures of probability distributions with the grid maximum likelihood method]. Avtomat. Rem. Contr. 71(3):455472.
 Dennis, J. E., and R. B. Schnabel. 1983. Numericalmeth ods for unconstrained optimization and nonlinear equations. Englewood Cliffs: PrenticeHall. 375 p.
[+] About this article
Title
THE POISSON THEOREM FOR BERNOULLI TRIALS WITH A RANDOM PROBABILITY OF SUCCESS AND A DISCRETE ANALOG OF THE WEIBULL DISTRIBUTION
Journal
Informatics and Applications
2016, Volume 10, Issue 4, pp 1120
Cover Date
20161230
DOI
10.14357/19922264160402
Print ISSN
19922264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
Bernoulli trials with a random probability of success; mixed binomial distribution; Poisson theorem; mixed Poisson distribution; Weibull distribution; PoissonWeibull distribution; mixed geometric distribution; EMalgorithm
Authors
V. Yu. Korolev , ,
A. Yu. Korchagin , ,
and A. I. Zeifman , ,
Author Affiliations
Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 152 Leninskiye Gory, GSP1, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center “Computer Sciences and Control” of the Russian
Academy of Sciences, 442 Vavilov Str.,Moscow 119333, Russian Federation
Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
ISEDT RAS, 56A Gorky Str., Vologda 16001, Russian Federation
