On a mixture representation of the conditional inactivity time of a coherent system

In this paper, we provide a mixture representation of the reliability function of the conditional inactivity time of a coherent system in terms of the reliability functions of conditional inactivity times of order statistics. Based on this representation, we then carry out stochastic comparisons between the conditional inactivity times of two coherent systems having either different structures or different component lifetimes. The results established here strengthen and extend some results known in the literature.