The number of photons returning from a target in a given time interval is well described by a negative-binomial distributed random variable. A photomultiplier tube (PMT) photon-counting detector is optimal for direct detection, and the number of detected-photon "electron pulses" produced is also negative-binomially distributed per time bin, with a reduced mean due to the device quantum efficiency. These tune distributed electron pulses are amplified and filtered by the preamplifier electronics prior to digitization and signal processing. The voltage output pulse per individual photo-electron event is known as the "impulse-response-function" of the detector and preamplifier. In this study we employ a typical analog preamplifier filter response, modeled as a Butterworth lowpass filter of order two, which filters a 200 ps wideband PMT input voltage pulse. The random summation of these lowpass voltage impulse-responses, as created by the negative-binomial photon arrival times and random photo-electron creation, is the classical electronic "shot-noise" random process. We derive numerically the voltage probability density function of this negative-binomial / impulse-response driven shot-noise random process following the stochastic process literature. We also show a technique to include PMT variations in gain, known as the "pulse height distribution," and to incorporate Gaussian baseline-noise voltage. Agreement with AMOR experiments is shown to be excellent. In addition, a Monte Carlo realization is presented, using the same impulse-response temporal shape, which also gives excellent agreement with AMOK data and with the analytical/numerical calculations.
