On Multivariate Distributions of Various Orders Obtained by Waiting for the r-th Success Run of Length k in Trials With Multiple Outcomes

Abstract

A sequence of independent trials with m + 1 mutually exclusive outcomes S, F 1 , F 2,…, F m is considered until the occurrence of the r-th nonoverlapping success run of length k, and the distributions of related random and the distributions of related random vectors are derived. First a new genesis scheme is established for the multivariate negative binomial distribution of order k type I, of Philippou, Antzoulakos and Tripsiannis (1988). It is shown that it is the distribution of the sum of two random vectors: the i-th component of the first one is the number of occurrences of F i and the i-th component of the second one is the total number of S’s which precede directly the occurrences of F i but do not belong to any success run of length k (1 ≤ i ≤ m). Furthermore, we obtain exact distributions of random vectors whose components are numbers of failures, non-overlapping runs of failures, successes, overlapping success runs of length l and success runs of length at least l. The majority of the above problems are also treated in the case of the generalized sequence of order k and corresponding results are established regarding the k and corresponding results are established regarding the multivariate extended negative binomial distribution of order k of Philippou and Antzoulakos (1990). The present paper generalizes several results of Aki and Hirano (1994, 1995).Keywords and phrasesMultivariate distributions of order k type Iextendednegative binomialgeometricgenesis schemesuccessfailure of type i run