On computing the distribution function of the sum of independent random variables

Abstract

We present an efficient approach to the determination of the convolution of random variables when their probability density functions are given as continuous functions over finite support. We determine that the best approach for large activity networks is to descretize the density function using Chebychev's points. Scope and purpose The convolution operation occurs frequently in the analysis of sums of independent random variables. Although the operation of convolution is elementary in its concept, it is rather onerous to implement on the computer for several reasons that are spelled out in the paper. It is our objective to present a computational scheme, based on discretization of continuous density functions, that is both easy to implement and mathematically “correct”, in the sense of satisfying equality of several moments of the approximation to the moments of the original density function. The approach presented in the paper can be easily programmed on the computer, and gives the desired convolution to any desired degree of accuracy.