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In the study of reliability of the technical systems, records model play an important role. Assume that the lower limit value of the first record is known, then we propose a definiton of the mean residual life of the future record. We predict mean residual of the future records under condition that the lower limit value of the mth record is known. Furthermore, we present generalization of the mean residual life of record based on the sequance of k-recordsand study its various properties. Finally some simulation results are provided.

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‎The equilibrium distributions have many applications in reliability theory, stochastic orderings and random processes. ‎The purpose of this paper is to introduce the equilibrium distributions and presents some results related to this issue. Some results are based on order statistics. ‎In this paper, ‎the generalized Pareto distributions are also analyzed and some basic relationships between the equilibrium distributions are presented.

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The Kumaraswamy distribution is a two-parameter distribution on the interval (0,1) that is very similar to beta distribution. This distribution is applicable to many natural phenomena whose outcomes have lower and upper bounds, such as the proportion of people from society who consume certain products in a given interval. 

In this paper, we introduce the family of Kumaraswamy-G distribution, and we detect its hazard rate function, reversible hazard rate function, mean residual life function, mean past life function, and the behavior of each of them. Also, we investigate the stochastic order concept of the family of Kumaraswamy-G distribution. Finally, in the form of a practical example, we analyze the suitability of the Kumaraswamy distribution for real data.

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This paper explore some extropy properties of the lifetime of coherent systems with the assumption that the lifetime distribution of system components are independent and identically distributed. The presented results are obtained using the concept of system signature. To this aim, we first provide an expression for extropy of the lifetime of coherent systems. Then, stochastic extropy comparisons are discussed for coherent systems under the condition that both systems have the same characteristic structure. In cases where the number of system components is large or the system has a complex structure, it is difficult or time-consuming to obtain the exact extropy value of the system lifetime. Therefore, bounds are also obtained for extropy. In addition, a new criterion for selecting a preferred system based on relative extropy is proposed, which considers the lifetime of the desired system closest to the parallel system.