Numerical inversion of probability generating functions

Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probability mass functions. However, numerical transform inversion has not been widely used. This lack of use seems to be due, at least in part, to good simple numerical inversion algorithms not being well known. To help remedy this situation, in this paper the authors present a version of the Fourier-series method for numerically inverting probability generating functions. They obtain a simple algorithm with a convenient error bound from the discrete Poisson summation formula. The same general approach applies to other transforms.