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Joint distributions of numbers of success runs of specified lengths in linear and circular sequences
Authors:Kiyoshi Inoue  Sigeo Aki
Institution:(1) The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106-8569 Tokyo, Japan;(2) Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, 560-8531 Toyonaka, Osaka, Japan;(3) Present address: Faculty of Economics, Seikei University, Kichijoji-Kitamachi, 180-8633 Musasino, Tokyo, Japan;(4) Present address: Department of Mathematics, Faculty of Engineering, Kansai University, 564-8680 Suita, Osaka, Japan
Abstract:In this paper, we study two joint distributions of the numbers of success runs of several lengths in a sequence ofn Bernoulli trials arranged on a line (linear sequence) or on a circle (circular sequence) based on four different enumeration schemes. We present formulae for the evaluation of the joint probability functions, the joint probability generating functions and the higher order moments of these distributions. Besides, the present work throws light on the relation between the joint distributions of the numbers of success runs in the circular and linear binomial model. We give further insights into the run-related problems arisen from the circular sequence. Some examples are given in order to illustrate our theoretical results. Our results have potential applications to other problems such as statistical run tests for randomness and reliability theory. This research was partially supported by the ISM Cooperative Research Program (2003-ISM.CRP-2007).
Keywords:Bernoulli trials  circular success runs  enumeration schemes  recursive scheme  circular binomial distribution of orderk            probability function  probability generating function  double generating function
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