Notes on the sum and maximum of independent exponentially distributed random variables with different scale parameters

TL;DR

This paper derives explicit density functions for the sum and maximum of independent, non-identically distributed exponential variables, revealing connections between order statistics, convolutions, and characteristic functions.

Contribution

It provides a simple, direct method to obtain the density functions for sums and maxima of non-i.i.d. exponential variables, linking order statistics and convolution representations.

Findings

Explicit density functions for sums and maxima are derived.

A connection between density functions and characteristic functions is established.

The approach simplifies analysis of non-i.i.d. exponential variables.

Abstract

We consider the distribution of the sum and the maximum of a collection of independent exponentially distributed random variables. The focus is laid on the explicit form of the density functions (pdf) of non-i.i.d. sequences. Those are recovered in a simple and direct way based on conditioning. A connection between the pdf and a representation of the convolution characteristic function as a linear combination of the single characteristic functions is drawn. It is demonstrated how the results on the pdf of order statistics and the convolution merge.