JOINT DISTRIBUTIONS OF NUMBERS OF SUCCESS-RUNS AND FAILURES UNTIL THE FIRST CONSECUTIVE k SUCCESSES

IN A BINARY SEQUENCE

N. BALAKRISHNAN

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

(Received February 1, 1996; revised July 1, 1996)

Abstract. Joint distributions of the numbers of failures, successes and success-runs of length less than k until the first consecutive k successes in a binary sequence were derived recently by Aki and Hirano (1995, Ann. Inst. Statist. Math., 47, 225–235). In this paper, we present an alternate derivation of these results and also use this approach to establish some additional results. Extensions of these results to binary sequences of order h are also presented.

Key words and phrases: Probability generating function, waiting time, binary sequence of order k, geometric distribution of order k.