The Number of Working Components in a Conditional Coherent Failed System

Coherent systems are very important in reliability,survival analysis and other life sciences. In this paper, we consider the number of working components in an (n-k+1)-out-of-n system, given that at least (n-m+1) components are working at time t, and the system has failed at time t. In this condition, we compute the probability that there are exactly i working components. First the reliability and several stochastic properties are obtained. Furthermore, we extend the results to general coherent systems with absolutely continuous and exchangeable components.