The inspection paradox with random time

When considering a delayed renewal process one may be interested in both, the renewal function and the expected length of the interarrival time that contains some fixed time t. In general, it is difficult to obtain explicit expressions for specific underlying distributions. Replacing t by a random variable T and using prior information about T, that is, assuming that T has some continuous NBU (NWU) distribution function G, bounds of the quantities are derived as well as representations, if T is exponentially distributed. As an implication an equation of Wald type is shown. The results can be applied to the analysis of control charts in quality control. Moreover, related bounds of a sample mean based on a random sample size are given and an elementary renewal reward theorem is stated.