Confidence intervals for efficiencies in particle physics experiments

We compute bias and variance of an efficiency estimator for a random processes, in which the success probability is constant but the number of trials is drawn from a Poisson distribution. The standard estimator, although being a non-linear function in this case, is unbiased. Compared to the case where the number of trials is fixed, the variance is increased or decreased depending on the expected number of trials. We further compute the variance for the case where the numbers of successes and failures have a variance which exceeds that of a Poisson process. This is the case, for example, when these numbers are obtained from a mixture of signal and background events in which the background is subtracted imperfectly. We compute generalised Wilson intervals based on these variances and study their coverage probability. We conclude that the standard Wilson interval is also suitable when the number of trials is Poisson distributed.