On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Abstract

Let \(X_1 ,X_2 ,\ldots ,X_n \) be a sequence of Markov Bernoulli trials (MBT) and \(\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })\) be a random vector where \(X_{n,k_i } \) represents the number of occurrences of success runs of length \(k_i \,( {i=1,2,\ldots ,r})\). In this paper the joint distribution of \(\underline{X}_n \) in the sequence of \(n\) MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least \(k\), overlapping runs, runs of exact length \(k\) and \(\ell \)-overlapping runs (i.e. \(\ell \)-overlapping counting scheme), \(0\le \ell \!:\!\!G\) system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed.