NUMBERS OF SUCCESS-RUNS OF SPECIFIED LENGTH UNTIL CERTAIN STOPPING TIME RULES AND GENERALIZED BINOMIAL DISTRIBUTIONS OF ORDER k

SIGEO AKI^{1} AND KATUOMI HIRANO^{2}

1Department of Informatics and Mathematical Science, Graduate School of Engineering Science,

Osaka University, 1-3 Machikaneyama-cho, Toyonaka, 560-8531, Japan

2The Institute of Statistical Mathematics, 4-6-7 Minani-Azabu, Minato-ku, Tokyo, 106-8569, Japan

(Received April 21, 1998; revised February 22, 1999)

Abstract. A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer μ less than the specified length of the runs, an enumeration scheme called μ-overlapping way of counting is defined. Let k and ℓ be positive integers satisfying ℓ < k. Based on independent Bernoulli trials, it is shown that the number of (ℓ−1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length ℓ follows the generalized binomial distribution of order (k −ℓ). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k −1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.

Key words and phrases: Binomial distribution of order k, Markov chain, probability generating function, stopping time, success-run, waiting time.