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

Generalizations of Runs and Patterns Distributions for Sequences of Binary Trials

“…The number of appearances of E in the sequence Z 1 , Z 2 , ..., Z n (n a fixed integer) will be denoted by X n . In a series of papers (Antzoulakos et al (2003), Dafnis et al (2012), Fu and Koutras (1994a), Koutras (2003), Koutras and Alexandrou (1995)) a Markov chain imbedding method was developed for the study of the exact distribution of enumerating random variables defined on sequences of binary (or multistate) trials. Since we are going to make use of this approach in the forthcoming sections, we shall first review the key features of it, bringing in at the same time all necessary terminology.…”

“…As already mentioned, the compound pattern E 1 was first introduced by Dafnis et al (2019), who studied the distribution of N (1) n,k,r , in the case of independent trials, motivated by a reliability evaluation problem in a consecutive-type system. Special cases of the compound patterns E 1 , E 2 have also been considered in Dafnis et al (2012), where the distributions of N (i) n,2,r (i = 1, 2) have been obtained when Z 1 , Z 2 , . .…”

“…For k = 2, some of the random variables introduced in the present article reduce to random variables enumerating patterns. Dafnis et al (2012) studied the distributions of N (i) n,2,r (i = 1, 2) employing a Markov chain imbedding technique. These distributions were also studied earlier by Sen and Goyal (2004), for r ≥ 1, by the aid of combinatorial methods.…”

On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

“…respectively. Some relevant contributions on the subject are the works of Sarkar et al [12], Sen and Goyal [13], Holst [4], Dafnis et al [14], Huffer and Sethuraman [15], and Makri and Psillakis [16]. Applications of constrained ( , ℓ) strings of the general ( ≤ ≤ ℓ) or the restricted forms ( ≤ ℓ, = , ≥ ) were found in information theory and data compression (see Zehavi and Wolf [17], Jacquet and Szpankowski [18], and Stefanov and Szpankowski [19]) in urn models, record models, and random permutations (see Chern et al [20], Joffe et al [21], Chern and Hwang [22], Holst [5,9,10,23], and Huffer et al [24]) in system reliability (see Eryilmaz and Zuo [25], Eryilmaz and Yalcin [6], and Makri [26]) and in biomedical engineering (see Dafnis and Philippou [27]).…”

On Limited Length Binary Strings with an Application in Statistical Control

“…The study, via M ;k,l , of constrained (k, l) strings where a 1 is followed by at least k and at most l 0s before the next 1 in a 0 -1 sequence covers as particular cases strings with zero run length at most equal (M n;0,l , d ≤ l), exactly equal (M n;k,k , d = k) and at least equal (M n;k,n-2 , d ≥ k) to a non-negative integer number. Some relevant contributions on the subject are the works of Sarkar et al [2], Sen and Goyal [3], Holst [4,5], Huffer et al [6], Eryilmaz and Zuo [7], Eryilmaz and Yalcin [8], Dafnis et al [9], and Makri and Psillakis [10]. Moreover, M n;0,0 is the Ling's [11] RV which counts overlapping runs of 1s of length 2 (with overlapping part of length at most 1), see e.g.…”