An approximate formula for the entropy of the negative binomial distribution

TL;DR

This paper derives an approximate formula for the Shannon entropy of the negative binomial distribution, useful for analyzing charged particle multiplicities with a deviation of about 20%.

Contribution

It provides a new approximate entropy formula for the NBD, addressing the lack of a closed-form expression.

Findings

Approximate entropy formula valid within 20% for extreme NBD parameters.

The formula simplifies entropy calculations for charged particle multiplicity studies.

Enhances theoretical understanding of the NBD's informational properties.

Abstract

Recent theoretical developments revived the interest in charged particle multiplicities and their wide-spread parametrization, the negative binomial distribution (NBD). The central observable of the studies is the Shannon entropy of the NBD. A closed form is not known, however, there are representations with special series and integrals. In this note, we will investigate one of these and give an approximate formula for the entropy that is valid up to $\sim$20\% deviation from the exact value for extreme values of the NBD parameters.