Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures |
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Authors: | Spiros?D?Dafnis Email author" target="_blank">Andreas?N?PhilippouEmail author Demetrios?L?Antzoulakos |
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Institution: | (1) Department of Mathematics, Anna University, Chennai, 600 025, India;(2) Department of Mathematics, R.M.K Engineering College, Chennai, 601 206, India |
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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 E1{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E2{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E3{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by Nn,k(i){ N_{n,k}^{(i)}} (respectively Mn,k(i){M_{n,k}^{(i)}}) the number of occurrences of the pattern Ei{\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 Tr,k(i){T_{r,k}^{(i)}} (resp. Wr,k(i)){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern Ei{\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 Nn,k(i){N_{n,k}^{(i)}}, Mn,k(i){M_{n,k}^{(i)}}, Tr,k(i){T_{r,k}^{(i)}} and Wr,k(i){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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Keywords: | |
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