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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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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.