The joint distribution of the number of occurrences of two interrelated Poisson processes

The focus of this paper is the distribution for the number of occurrences of two independent but interrelated Poisson processes, to be named the joint imputed Poisson distribution. From its inception, the research was motivated by a problem facing the electric power industry. Power surges, which are observable, represent occurrences of the imputing process. Circuit breaker breakdowns, which cannot be seen taking place, constitute occurrences of the imputed process. Though not observable, at each occurrence of the imputing process, knowledge about the imputed process is obtained. Occurrences of the imputed process are revealed only as the result of periodic inspection, or with far greater cost at the subsequent occurrence of the imputing process. A comprehensive treatment of the joint imputed Poisson distribution for the number of occurrences of the two processes over a fixed time period was presented by Cuffe. Because these probabilistic results had to be derived from first principles, a lengthy, rigorous mathematical exposition is not appropriate in this forum. Rather, we concentrate here on a presentation of results with only highlights of their proofs.