ON A WAITING TIME DISTRIBUTION IN A SEQUENCE OF BERNOULLI TRIALS

M. V. KOUTRAS

Department of Mathematics, University of Athens, Panepistemiopolis, 15784, Greece

(Received June 27, 1995; revised November 6, 1995)

Abstract. In the present article we investigate the exact distribution of the waiting time for the r-th non-overlapping appearance of a pair of successes separated by at most k − 2 failures (k ≥ 2) in a sequence of independent and identically distributed (iid) Bernoulli trials. Formulae are provided for the probability distribution function, probability generating function and moments and some asymptotic results are discussed. Expressions in terms of certain generalised Fibonacci numbers and polynomials are also included.

Key words and phrases: Waiting times, success runs, scans, distributions of order k, Fibonacci numbers, unimodality, Markov dependence.