SOONER AND LATER WAITING TIME PROBLEMS FOR RUNS IN MARKOV DEPENDENT BIVARIATE TRIALS

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

1Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan 2The 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.