Time-interval density distributions of accepted events in a HPGe γ-ray detection set-up are measured with a time-interval digitiser. In particular, the effect of pulse-pileup rejection is investigated. Experimental data are obtained with two types of shaping amplifiers: a classical amplifier with semi-Gaussian pulse shaping and a gated-integrator amplifier. A theoretical model is developed to predict typical time-interval density distributions for stationary Poisson processes passing through a detector with count loss by pulse-pileup rejection. Good agreement is obtained between theoretical, measured and simulated time-interval spectra. It is found that, when counting is affected by pileup rejection, the true incoming count rate cannot simply be determined by fitting an exponential to the time-interval distributions. From the Laplace transform of the interval-density distribution, expressions are derived for the expectation value and the variance of the counts. Good agreement is found with experimental counting statistics for different system configurations, as well as with data from computer simulations.