Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales
Abstract
It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of a function by a simple convolution. This allows efficient numerical computations of moment-generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it in high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwis difficult to handle. The computational efficiency compared to other methods is demonstrated for a M/G/1 queuing problem.