In view of the growing importance of reversed hazard rate (RHR) in reliability analysis and stochastic modeling, we have considered different implicative relationships with respect to the monotonic behavior of RHR. In that context, a few characterizing properties have also been presented based on expected inactivity time.

A new probability distribution, the xgamma distribution, is proposed and studied. The distribution is generated as a special finite mixture of exponential and gamma distributions and hence the name proposed. Various mathematical, structural, and survival properties of the xgamma distribution are derived, and it is found that in many cases the xgamma has more flexibility than the exponential distribution. To evaluate the comparative behavior, stochastic ordering of the distribution is studied. To estimate the model parameter, the method of moment and the method of maximum likelihood estimation are proposed. A simulation algorithm to generate random samples from the xgamma distribution is indicated along with a simulation study. A real life dataset on the remission times of patients receiving an analgesic is analyzed, and it is found that the xgamma model provides better fit to the data as compared to the exponential model.

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A B S T R A C TAn existing one-parameter probability distribution can be very well generalized by adding an extra parameter in it and, in turn, the two-parameter family of distributions, thus obtained, provides added flexibility in modeling real life data. In this article, we propose and study a two-parameter generalization of xgamma distribution [1] and utilize it in modeling time-to-event data sets. Along with the different structural and distributional properties of the proposed two-parameter xgamma distribution, we concentrate in studying useful survival and reliability properties, such as hazard rate, reversed hazard rate, stress-strength reliability etc. Two methods of estimation, viz. maximum likelihood and method of moments, are been suggested for estimating unknown parameters. Distributions of order statistics, stochastic order relationships are investigated for the proposed model. A Monte-Carlo simulation study is carried out to observe the trends in estimation process. Two real life time-to-event data sets are analyzed and the proposed model is compared with some other two-parameter lifetime models in the literature.This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). The TPXG along with its alternative form is introduced in section 2. The moments and related measures are studied in section 3. Incomplete moments are utilized in studying famous inequality curves and different entropy measures are studied in sections 4 and 5, respectively.

Mostly discrete distributions have been derived and studied independently of any continuos counter part. But sometimes we find that some properties of a continuous distribution, if applied in the discrete support, also characterises a discrete distribution. In these situations we call the discrete distribution as the discretized version of the continuous distribution. For example, Geometric distribution is the discrete version of the Exponential distribution. Many authors have discretized continuous random variables by using different properties of the continuous distributions viz. distribution function, moments, characterizing properties etc. They have also shown various applications of these discretized random variables. For example, in a complex system, the system response function is a random variable due to random nature of the component lives. But for the complexity of the system, generally analytical expressions for the response function are difficult to obtain. In such situations discretization of the continuous variables helps us to study the same. Here we have discretized the Weibull distribution by using its failure rate function. Application of the discretization has been demonstrated in approximating the reliability of Solid-shaft, a well-known engineering item. Numerical study shows that the proposed discretization gives a better approximation of the system reliability over the methods of discrete concentration, which is based on survival function, moment equalization and frequency curve equalization.