The observed hazard and multicomponent systems

Let X denote the life of some system. We define the observed hazard rate at time t , call it R ( t ), as the instantaneous probability (density) of failure of X at time t given survival up to t and given a complete description of the system state at t. We conjecture that the total observed hazard—namely, σo X R ( t ) dt —is an exponential random variable with mean 1 and verify this for the special case when X is the distribution of system life of an n component system having an arbitrary monotone structure function.