A probabilistic analogue of the mean value theorem and its applications to reliability theory

In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whin [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in Various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of Lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.