The moments and central moments of a compound distribution

The compound distribution is of interest for the study of production/inventory problems, since it provides a flexible description of the stochastic properties of the system. However, due to the difficulties involved in obtaining analytical results for the compound distribution, studies are usually limited to searching for a good approximation by replacing a more complex model with a simpler one applying only the first few moments as parameters. This paper presents general closed form formulae for the moments and central moments of any order of a compound distribution made up of non-negative stochastic variables. The Laplace and z-transform methods play an important role in this study. The importance of taking into consideration higher-order moments, when computing a safety factor for inventory control, is illustrated in a numerical example.