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A new over-dispersed discrete probability model is introduced, by compounding the Poisson distribution with the weighted Ishita distribution. The statistical properties of the newly introduced distribution have been derived and discussed. Parameter estimation has been done with the application of the maximum likelihood method of estimation, followed by the Monte Carlo simulation procedure to examine the suitability of the ML estimators. In order to verify the applicability of the proposed distribution, a real-life set of data from the medical field has been analysed for modeling a count dataset representing epileptic seizure counts.

In this paper, we shall introduce a new class of generalized complementary compound lifetime distributions which is obtained by compounding generalized Lindley distribution with power series distribution. This new family of continuous lifetime distributions so obtained will be called Complementary Generalized Lindley Power Series (CGLPS) distribution. The proposed class of distribution contains several lifetime distributions as its special cases that are very flexible to accommodate different types of data sets since the probability density function and hazard rate can take up different forms such as increasing, decreasing and upside down bathtub shapes which have been shown through graphs for some selected values of parameters and the potentiality proposed class has been tested statistically by using it to model some real life data set.

In a system with standby redundancy, there are a number of components only one of which works at a time and the other remain as standbys. When an impact of stress exceeds the strength of the active component, for the first time, it fails and another from standbys, if there is any, is activated and faces the impact of stresses, not necessarily identical as faced by the preceding component and the system fails when all the components have failed. Sriwastav and Kakaty (1981) assumed that the components stress-strengths are similarly distributed. However, in general the stress distributions will be different from the strength distributions not only in parameter values but also in forms, because stresses are independent of strengths and the two are governed by different physical conditions. Assume the components in the system for both stress and strength are independent and follow different probability distributions viz. Exponential, Gamma, Lindley. Different conditions for stress and strength were considered. Under these assumptions the reliabilities of the system have been obtained with the help of the particular forms of density functions of n-standby system when all stress-strengths are random variables. The expressions for the marginal reliabilities R(1), R(2), R(3) etc. have been obtained based on its stress-strength models. Results obtained by J. Gogoi and M. Bohra are particular case presentations.

Discrete distributions have played an important role in the reliability theory. In order to obtain Bayes estimators, researchers have adopted various conventional techniques. Generalizing the results of Maiti (1995), Chaturvadi and Tomer (2002) dealt with the problem of estimating P{X1, X2, …, Xk ≤ Y}, where random variables X and Y were assumed to follow a negative binomial distribution. Agit et al. obtained Bayesian estimates of the reliability functions and P{X1, X2, …, Xk ≤ Y} considering X and Y following binomial and Poisson distributions. The reliability function of the generalized Poisson and generalized geometric distribution is investigated. The expression for P{X1, X2, …, Xk ≤ Y} was obtained with X’s and Y following a Poisson distribution and some particular cases are shown.