A bivariate point process connected with electronic counters

Consider three independent Poisson processes of point events of rates $\lambda $$_{1}$, $\lambda $$_{2}$ and $\lambda $$_{12}$. There are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters have constant dead-time, i.e. following the recording of an event on a counter no further event can be recorded on that counter until the appropriate constant time has elapsed. Two ways of estimating $\lambda $$_{12}$ are via a coincidence rate, i.e. the rate of occurrence of pairs of events separated by less than a suitable small tolerance, and via the covariance of the numbers of events recorded on the two counters in a suitable time period. The theoretical values of these quantities are calculated allowing for dead-time. The techniques used illustrate the study of bivariate point processes.