On the density for sums of independent exponential, Erlang and gamma variates

TL;DR

This paper introduces a unified method using divided differences to derive closed-form densities for sums of independent exponential, Erlang, and gamma variables, offering new insights and approaches in convolution analysis.

Contribution

It presents a novel unified framework based on divided differences for deriving densities of sums of these distributions, including a new approach for gamma variates using fractional calculus.

Findings

Unified divided difference approach for exponential, Erlang, gamma sums

New method for gamma sum densities via fractional calculus

Provides closed-form formulas for convolutions of these distributions

Abstract

This paper re-examines the density for sums of independent exponential, Erlang and gamma random variables. By using a divided difference perspective, the paper provides a unified approach to finding closed-form formulae for such convolutions. In particular, the divided difference perspective for sums of Erlang variates suggests a new approach to finding the density for sums of independent gamma variates using fractional calculus.