DISTRIBUTIONS OF NUMBERS OF FAILURES AND SUCCESSES UNTIL THE FIRST CONSECUTIVE k SUCCESSES

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

1Department of Mathematical Science, Faculty of Engineering Science, Osaka University,

Machikaneyama-cho, Toyonaka, Osaka 560, Japan 2The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

(Received March 15, 1993; revised June 21, 1993)

Abstract. Exact distributions of the numbers of failures, successes and successes with indices no less than l (1 ≤ l ≤ k − 1) until the first consecutive k successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Markov chain and a binary sequence of order k. The number of failures until the first consecutive k successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}- sequence is an iid sequence or a Markov chain, the distribution of the number of successes with indices no less than l is shown to be a shifted geometric distribution of order k −l. When the {0, 1}-sequence is a binary sequence of order k, the corresponding number follows a shifted version of an extended geometric distribution of order k − l.

Key words and phrases: Geometric distribution, discrete distributions, Markov chain, waiting time, geometric distribution of order k, iid sequence, binary sequence of order k, inverse sampling.