Given a coherent system, let Z be the age of the machine at breakdown, and I the set of parts failed by time Z. Assume that the component lifetimes are independent. Assume further that the lifetime distributions are mutually absolutely continuous and that each possesses a single positive atom at the common essential infimum. We prove that the joint distribution of (Z, I) identifies the lifetime distribution of each part if and only if there is at most one component belonging to all cut sets. If we relax the mutual absolute continuity assumption by allowing isolated intervals of constancy, then a necessary and sufficient condition for identifiability is that no two parts be in parallel.

Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovič iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.