Abstract
Let Z 1, Z 2, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z t = 1) and q = Pr(Z t = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns \({\mathcal{E}_{1}}\): two successes are separated by at most k−2 failures, \({\mathcal{E}_{2}}\): two successes are separated by exactly k −2 failures, and \({\mathcal{E}_{3}}\) : two successes are separated by at least k − 2 failures. Denote by \({ N_{n,k}^{(i)}}\) (respectively \({M_{n,k}^{(i)}}\)) the number of occurrences of the pattern \({\mathcal{E}_{i}}\) , i = 1, 2, 3, in Z 1, Z 2, . . . , Z n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let \({T_{r,k}^{(i)}}\) (resp. \({W_{r,k}^{(i)})}\) be the waiting time for the r − th occurrence of the pattern \({\mathcal{E}_{i}}\), i = 1, 2, 3, in Z 1, Z 2, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of \({N_{n,k}^{(i)}}\), \({M_{n,k}^{(i)}}\), \({T_{r,k}^{(i)}}\) and \({W_{r,k}^{(i)}}\) (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.
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Dafnis, S.D., Philippou, A.N. & Antzoulakos, D.L. Distributions of patterns of two successes separated by a string of k-2 failures. Stat Papers 53, 323–344 (2012). https://doi.org/10.1007/s00362-010-0340-7
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DOI: https://doi.org/10.1007/s00362-010-0340-7