Joint distributions of numbers of success-runs and failures until the first consecutivek successes |
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Authors: | Sigeo Aki Katuomi Hirano |
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Affiliation: | (1) Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Machikaneyama-cho, 560 Toyonaka, Osaka, Japan;(2) The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106 Tokyo, Japan |
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Abstract: | Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.This research was partially supported by the ISM Cooperative Research Program (93-ISM-CRP-8). |
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Keywords: | Probability generating function geometric distribution discrete distributions Markov chain waiting time geometric distribution of orderk binary sequence of orderk |
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