In the present work we study the distribution of a random sum of random variables which is related to a binary scan statistic for Markov dependent trials. The motivation of the model studied herein stems from several areas of applied science such as actuarial science, financial risk management, quality control and reliability, educational psychology, engineering, etc.

Let us consider a sequence of binary success/failure trials and denote by T_k the waiting time for the first occurrence of two successes separated by at most k failures, where k≥0 is any integer. Let also Y₁,Y₂,… be a sequence of independent and identically distributed (i.i.d) discrete random variables, independent of T_k. In the present article we develop some results for the distribution of the compound random variable and illustrate how these results can be profitably used to study models pertaining to actuarial science and financial risk management practice.

In ottoman society, opportunity of acquiring luxurious elements stands out as an upper-class practice. In this study Ottoman state archive documents was used to examine the issue in context of house, and the data obtained were assessed within the context of their physical properties and urban locations. Consequently, the diversity and the prevalence of house components was determined and their purposes are shown through period documents.

In this paper, we generalize geometric and binomial distributions of order k to q-geometric and q-binomial distributions of order k using Bernoulli trials with a geometrically varying success probability. In particular, we derive expressions for the probability mass functions of these distributions. For q=1, these distributions reduce to geometric and binomial distributions of order k which have been extensively studied in the literature.

In a sequence of n binary trials, distribution of the random variable M_{n,k}, denoting the number of overlapping success runs of length exactly k, is called Ling’s binomial distribution or Type II binomial distribution of order k. In this paper, we generalize Ling’s binomial distribution to Ling’s q-binomial distribution using Bernoulli trials with a geometrically varying success probability. An expression for the probability mass function of this distribution is derived. For q=1, this distribution reduces to Ling’s binomial distribution.

Start-up demonstration testing is an effective method for illustrating the reliability of a unit before purchasing it. The test consists of starting-up the unit, and observing the outcomes, either success or failure. According to the total successes total failures (TSTF) test procedure, a unit under test is accepted when a specified number of successes is observed before a specified number of failures; otherwise, the unit is rejected. We study the TSTF procedure for dependent start-ups where the outcome of the present start-up depends on the total number of successful start-ups so far. The main characteristics of the TSTF test are obtained under this previous-sum dependent model, and numerical illustrations are presented.

This paper is concerned with the mean, minimum and maximum distances between two successive failures in a binary sequence consisting of Markov dependent elements. These random variables are potentially useful for the analysis of the frequency of critical events occurring in certain stochastic processes. Exact distributions of these random variables are derived via combinatorial techniques and illustrative numerical results are presented.

Let {X_i}, i≥1 be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in {X_i}, i≥1. Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models.