On computing distributions of products of non-negative independent random variables

TL;DR

This paper introduces a novel functional representation for PDFs of non-negative random variables using exponential mixtures, enabling efficient computation of distributions of sums, products, or quotients with high accuracy.

Contribution

The authors develop a new approximate PDF representation and a fast numerical algorithm for computing distributions of combined non-negative random variables.

Findings

Accurate approximation of PDFs using exponential mixtures.

Efficient computation of sums, products, or quotients of non-negative variables.

Validated approach with multiple numerical examples.

Abstract

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.