On the Use of Generating Functions and Laplace Transforms in Applied Probability Theory

Summary This article is concerned with the application of probability generating functions and Laplace transforms to the solution of specific probability problems. In particular, it presents the interpretation (or collective marks) method of van Dantzig and Runnenburg. With this method, generating functions and Laplace transforms can be interpreted as probabilities and these can often be obtained directly, without having first to derive a recursion formula or a differential equation. It is demonstrated how this method can be used to derive the binomial, the Poisson and the compound Poisson distributions, and it is also applied to the thinning of renewal streams and to some stochastic service systems.