For a given system observed at timet, the past entropy serves as an uncertainty measure about the past life-time of the distribution. We consider a coherent system in which there arencomponents that have all failed at timet. To assess the predictability of the life-time of such a system, we use the signature vector to determine the entropy of its past life-time. We explore various analytical results, including expressions, bounds, and order properties, for this measure. Our results provide valuable insight into the predictability of the coherent system’s life-time, which may be useful in a number of practical applications.