Computation of counting distributions arising from a single-stage multiplicative process

The cumulative distribution of the number of secondary electrons in a single-stage photomultiplier is calculated by numerically integrating the inversion integral for its probability generating function along a suitably chosen contour. A residue series applicable in certain cases is also presented. Saddlepoint approximations to the contour integral are described, which are the more accurate, the greater the numbers of secondaries. Recurrent relations are developed for computing values of the distribution for purposes of comparison. Computation of the Neyman Type-A distribution is treated as a limiting case.