The reliability of composite system (series, parallel) is improved by (i) reduction method, and by (ii) hot duplication, considering the systems survivor function. Related survivor equivalence functions and pointwise survivor equivalence factors are derived in all cases when the components lifetime distribution follow the gamma–Weibull distribution introduced recently by Leipnik and Pearce.

A generalization of the log-logistic distribution so-called the transmuted log-logistic distribution is proposed and studied. Various structural properties including explicit expressions for the moments, quantiles, mean deviations of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. We hope that the new distribution proposed here will serve as an alternative model to the other models which are available in the literature for modeling positive real data in many areas.

Recently, Alkarni and Oraby (2012) obtained general forms for some properties of the compound class of Poisson and lifetime (PL) distributions. In this paper, we obtain some general forms for joint density, cumulative distribution, and survival functions of the bivariate case of PL class. Its conditional distributions are also studied. In addition, the compound class of geometric and lifetime distributions as well as its mixed bivariate case are discussed. For this class some conditional probabilities useful for reliability, biological survey, and engineering are also studied. Our class contains several new mixed bivariate distributions in special cases.

We have proposed measures of cumulative entropy and dynamic cumulative entropy based on order statistics and derived a characterization result that dynamic cumulative entropy of the i th order statistics determines the distribution function uniquely. Some properties of the measures proposed have also been studied.

Markov chains are useful to model various complex systems. In numerous situations, the underlying Markov chain is subject to changes. For example, states may be added or deleted and transition probabilities perturbed. It is therefore, necessary to ensure the robustness of the system and to estimate the resulting deviation in the characteristics. In this paper we study the sensitivity of finite Markov chains subject to changes in their state space and propose updating formulas and perturbation bounds.

An extended form of beta distribution by Al-Saleh and Agarwal, is further extended which has an additional two shape parameters k and l. Introduction of new shape parameters help to express extended beta distribution not only as a mixture of distributions, but also provides extra flexibility to the density function over the interval [0,1]. Certain statistical properties such as the r-th moment are defined explicitly. Some of the shapes of family of the densities are also illustrated for different k and l so that it may help the Bayesians to approximate a wide range of prior beliefs among the members of the suggested extended family. The Bayesian analysis for the posterior of an uncertain parameter for the Bernoulli process using extended beta prior is also considered with an application of mortality rates in 12 hospitals performing surgery on babies.

We consider the asymptotic behavior of the ?nal size of a multitype collective Reed-Frost process. This type of models was introduced by and include most known epidemic models of the type SIR (Susceptible, Infected, Removed) as special cases. Under certain conditions, we show that, when the initial number of susceptible is very large and the initial number of infected individuals is ?nite, the infection process behaves as a multitype Galton-Watson process. This fact is proved using a simple argument based on Bernstein polynomials. We use this result to study the ?nal size of the epidemic.

Distributional properties of two non-adjacent dual generalized order statistics have been used to characterize distributions. Further, one sided contraction and dilation for the dual generalized order statistics are discussed and then the results are deduced for generalized order statistics, order statistics and adjacent dual generalized order statistics and generalized order statistics.