Distributions of patterns of two successes separated by a string of k-2 failures

• Published in: Statistical papers (Berlin, Germany) Germany), 2012-05, Vol.53 (2), p.323-344
• Main Authors: Dafnis, Spiros D., Philippou, Andreas N., Antzoulakos, Demetrios L.
• Format: Article
• Language: English
• Subjects: Counting; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Failure; Finance; Insurance; Integers; Management; Markov analysis; Mathematics and Statistics; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Random variables; Recursive; Regular Article; Statistics; Statistics for Business; Strings; Studies; Success
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-010-0340-7

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 : two successes are separated by at most k −2 failures, : two successes are separated by exactly k −2 failures, and : two successes are separated by at least k − 2 failures. Denote by (respectively ) the number of occurrences of the pattern , 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 (resp. be the waiting time for the r − th occurrence of the pattern , 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 , , and ( i  = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.