Uncertainty in Poisson event counts and exposure time in rate estimation

In quantifying a plant-specific Poisson event occurrence rate λ in Probabilistic Risk Assessment, it is sometimes the case that either the corresponding Poisson exposure time t or the observed number of events x (or both) are uncertain. We present several methods which account for uncertainties in both x and t when using Bayesian methods to estimate λ. A gamma prior distribution on λ is considered. While the methods formally require numerical integration, a computationally convenient approximation is provided to implement them in practice. A numerical example concerning the rate of failure to operate of the high pressure coolant injection system of commercial boiling water reactors is used to illustrate the methods.