  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
after-pulses.
  This study investigate 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
after-pulses. Expressions for statistical moments of any order are obtained on
the basis of the generalized Poisson distribution (GPD). 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
simple experiment are considered. The results of the experimental verification
of the theoretical expression confirm the need to account for the non-ideal
nature of detectors in almost all similar measurements.
