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

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.