Formulae and recursions for the joint distribution of success runs of several lengths
Consider a sequence of n independent Bernoulli trials with the j-th trial having probability Pj of success, 1 ≤ j ≤ n. Let M(n, K) and N(n,K) denote, respectively, the r-dimensional random variables (M(n, k1), ..., M(n,kr)) and (N (n, k1),..., N(n, kr)), where K = (k1, k2,..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.
Annals of the Institute of Statistical Mathematics
Formulae and recursions for the joint distribution of success runs of several lengths.
Annals of the Institute of Statistical Mathematics,
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/8209