SUCCESS RUNS OF LENGTH k IN MARKOV DEPENDENT TRIALS

S. G. MOHANTY

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

(Received June 16, 1993; revised December 9, 1993)

Abstract. The geometric type and inverse Polýa-Eggenberger type distributions of waiting time for success runs of length k in two-state Markov dependent trials are derived by using the probability generating function method and the combinatorial method. The second is related to the minimal sufficient partition of the sample space. The first two moments of the geometric type distribution are obtained. Generalizations to ballot type probabilities of which negative binomial probabilities are special cases are considered. Since the probabilities do not form a proper distribution, a modification is introduced and new distributions of order k for Markov dependent trials are developed.

Key words and phrases: Ballot problem, success runs, Markov dependent trials, discrete distributions.