Demir Atalay ve Zeybek, 2013

Journal of Statistical Planning and Inference, 2013

Statistical Papers , 2005

The number of l-overlapping success runs of length k in n trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number of l-overlapping success runs of length k in n Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of order k.

Proc 20th Panhel Stat Conf (2007), 479-487, Nicosia, Cyprus, 2007

The numbers of l-overlapping success runs of length k in n two state (success-failure) trials arranged on a line or on a circle, as well as a waiting time random variable associated with the l-overlapping enumerative scheme, are studied. The probability mass functions are derived by a simple combinatorial approach, through a model of allocation of balls into cells, and they are given in closed formulae in terms of certain combinatorial numbers. The study, is developed first for Bernoulli trials, and then it is generalized to the Polya-Eggenberger sampling scheme.

Microelectronics Reliability, 1996

A linear (circular) m-consecutive-k-out-of-n:F system, consists of n components ordered on a line (on a circle). The system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Four theorems are given. Theorem 1 is an exact formula of the reliability of the linear system given in terms of multinomial coefficients. Theorem 2 is an exact formula of the reliability of the linear system in terms of binomial coefficients. Theorems 3 and 4 are exact formulas of the reliability of the circular system, given in terms of multinomial and binomial coefficients, respectively.

Journal of Statistical Planning and Inference, 2007

The shortest and the longest length of success runs statistics in binary sequences are considered. The sequences are arranged on a line or on a circle. Exact probabilities of these statistics are derived, both in closed formulae via combinatorial analysis, as well as recursively. Furthermore, their joint probability distribution function and cumulative distribution function are obtained. The results are developed first for Bernoulli trials (i.i.d. binary sequences), and then they are generalized to the Polya-Eggenberger sampling scheme. For the latter case, the length of the longest success run is related to other success runs statistics and to reliability of consecutive systems.

Annals of the Institute of Statistical Mathematics, 2003

In the present paper, we study the distribution of a statistic utilizing the runs length of "reasonably long" series of alike elements (success runs) in a sequence of binary trials. More specifically, we are looking at the sum of exact lengths of subsequences (strings) consisting of k or more consecutive successes (k is a given positive integer). The investigation of the statistic of interest is accomplished by exploiting an appropriate generalization of the Markov chain embedding technique introduced by Fu and , Y. Amer. Statist. Assoc., 89, 1050-1058 and Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743-766). In addition, we explore the conditional distribution of the same statistic, given the number of successes and establish statistical tests for the detection of the null hypothesis of randomness versus the alternative hypothesis of systematic clustering of successes in a sequence of binary outcomes.

International Journal of Mathematics and Mathematical Sciences, 2003

In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of orderk, typeI, which extends to distributions of orderk, the generalized negative binomial distribution of Jain and Consul (1971), and includes as a special case the negative binomial distribution of orderk, typeI, of Philippou et al. (1983). This new distribution gives rise in the limit to generalized logarithmic and Borel-Tanner distributions and, by compounding, to the generalized Pólya distribution of the same order and type. Limiting cases are considered and an application to observed data is presented.

Journal of Probability and Statistics

In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of orderk, expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials withr≥1occurrences of success runs of lengthkwithl-overlapping. The casel<0, where there must be a gap of at leastltrials between success runs, is also studied. Next we treat the distribution of the waiting time for therthnonoverlapping appearance of a pair of successes separated by at mostk-2failures (k≥2).

Statistics & Probability Letters, 1996

New simple formulae for some probability distributions of success runs in Bernoulli trials are found by using the classical definition of run. These expressions contain only one summation of ordinary binomial coefficients and thus allow a faster and efficient computation.

Computers & Mathematics with Applications, 2011

Consider a sequence of n Bernoulli (Success-Failure or 1-0) trials. The exact and limiting distribution of the random variable E n,k denoting the number of success runs of a fixed length k, 1 ≤ k ≤ n, is derived along with its mean and variance. An associated waiting time is examined as well. The exact distribution is given in terms of binomial coefficients and an extension of it covering exchangeable sequences is also discussed. Limiting distributions of E n,k are obtained using Poisson and normal approximations. The exact mean and variance of E n,k which are given in explicit forms are also used to derive bounds and an additional approximation of the distribution of E n,k . Numbers, associated with E n,k and related random variables, counting binary strings and runs of 1's useful in applications of computer science are provided. The overall study is illustrated by an extensive numerical experimentation.