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On the Waiting Time for the First Success Run

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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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka, 564-8680, Japan

    Sigeo Aki

  2. The Institute of Statistical Mathematics, Research Organization of Information and Systems, 4-6-7 Minami-Azabu, Minato-ku, Tokyo, 106-8569, Japan

    Katuomi Hirano

  3. Department of Statistical Science, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies, Tokyo, Japan

    Katuomi Hirano

Authors
  1. Sigeo Aki
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  2. Katuomi Hirano
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Corresponding author

Correspondence to Katuomi Hirano.

Additional information

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)

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Aki, S., Hirano, K. On the Waiting Time for the First Success Run. AISM 59, 597–602 (2007). https://doi.org/10.1007/s10463-006-0059-3

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  • DOI: https://doi.org/10.1007/s10463-006-0059-3

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