DISTRIBUTIONS OF THE NUMBERS OF FAILURES AND SUCCESSES IN A WAITING TIME PROBLEM

STATHIS CHADJICONSTANTINIDIS AND MARKOS V. KOUTRAS

Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece

(Received February 8, 1999; revised July 8, 1999)

Abstract. In this article we consider infinite sequences of Bernoulli trials and study the exact and asymptotic distribution of the number of failures and the number of successes observed before the r-th appearance of a pair of successes separated by a pre-specified number of failures. Several formulae are provided for the probability mass function, probability generating function and moments of the distribution along with some asymptotic results and a Poisson limit theorem. A number of interesting applications in various areas of applied science are also discussed.

Key words and phrases: Waiting times, scan statistics, success runs, unimodality, partial fraction expansions, Poisson limit theorems.