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On the Waiting Time for the First Success Run |
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| Authors: | Sigeo Aki Katuomi Hirano |
| Institution: | (1) Department of Mathematics, Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita Osaka, 564-8680, Japan;(2) The Institute of Statistical Mathematics, Research Organization of Information and Systems, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan;(3) Department of Statistical Science, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies, Tokyo, Japan |
| Abstract: | Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained. This research was partially supported by the ISM Cooperative Research Program(2004-ISM-CRP-2006) and by a Grant-in-Aid for Scientific Research (C) of the JSPI (Grant Number 16500183) |
| Keywords: | Geometric distribution of order k Probability generating function Conditional expectation Markov chain Run Discrete distribution |
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