Many physical experiments require analysis of the statistics of fluctuating radiation. In the case of an ideal single-photon detector, the contribution of photon noise to the statistics of the registered signal has been thoroughly examined. However, practical photon counters have a dead time, leading to miscounting of certain true events, and sometimes the counters generate false afterpulses. This study investigates the impact of these two effects, and it presents the theoretical relations between the statistical moments of the radiation and the registered counts while also accounting for dead time and the probability of afterpulses. Expressions for statistical moments of any order are obtained on the basis of the generalized Poisson distribution. For counters with paralyzable dead time, alternative relations for the mean and variance are derived using generally accepted formulas. As an example, the measurements of stellar scintillation and the result of a simple experiment are considered. The results of the experimental verification of the theoretical expression confirm the need to account for the nonideal nature of detectors in almost all similar measurements.
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