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SOONER AND LATER WAITING TIME PROBLEMS FOR

RUNS IN MARKOV DEPENDENT BIVARIATE TRIALS

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SIGEO AKI^{1} AND KATUOMI HIRANO^{2}

^{1} *Department of Informatics and Mathematical Science,*

Graduate School of Engineering Science, Osaka University,

1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan

^{2} *The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu,*

Minato-ku, Tokyo 106-8569, Japan
(Received July 7, 1997; revised March 2, 1998)

**Abstract.**
In this paper we study exact
distributions of sooner and later waiting times for runs in
Markov dependent bivariate trials. We give systems of linear
equations with respect to conditional probability generating
functions of the waiting times. By considering bivariate
trials, we can treat very general and practical waiting time
problems for runs of two events which are not necessarily
mutually exclusive. Numerical examples are also given in
order to illustrate the feasibility of our results.

*Key words and phrases*:
Discrete distributions,
probability generating function, run, waiting time problem.

**Source**
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