PROBABILITY DISTRIBUTION FUNCTIONS OF SUCCESSION QUOTAS IN THE CASE OF MARKOV DEPENDENT TRIALS

DEMETRIOS L. ANTZOULAKOS^{1} AND ANDREAS N. PHILIPPOU^{2}

1Department of Economics, University of Crete, Rethimno 74100, Greece 2Department of Mathematics, University of Patras, Patras 26100, Greece

(Received February 1, 1996; revised June 10, 1996)

Abstract. The exact probability distribution functions (pdf’s) of the sooner and later waiting time random variables (rv’s) for the succession quota problem are derived presently in the case of Markov dependent trials. This is done by means of combinatorial arguments. The probability generating functions (pgf’s) of these rv’s are then obtained by means of enumerating generating functions (enumerators). Obvious modifications of the proofs provide analogous results for the occurrence of frequency quotas and such a result is established regarding the pdf of a frequency and succession quotas rv. Longest success and failure runs are also considered and their joint cumulative distribution function (cdf) is obtained.

Key words and phrases: Sooner and later problems, frequency quota, succession quota, Markov dependent trials, probability distribution function, probability generating function, cumulative distribution function, binomial expressions, longest run, enumerator.