Entropies of deformed binomial distributions

TL;DR

This paper investigates the asymptotic properties and entropy measures of generalized binomial distributions derived from different polynomial and exponential functions, revealing diverse entropy extensivity behaviors.

Contribution

It introduces three specific examples of deformed binomial distributions and analyzes their entropy properties, including a probabilistic model for the Abel polynomial case.

Findings

The q-exponential derived distribution exhibits extensive Boltzmann-Gibbs entropy.

The Hermite polynomial-based distribution also shows extensive Boltzmann-Gibbs entropy.

The Abel polynomial distribution uniquely exhibits extensive Renyi entropy.

Abstract

Asymptotic behavior (with respect to the number of trials) of symmetric generalizations of binomial distributions and their related entropies are studied through three examples. The first one derives from the q-exponential as a generating function. The second one involves the modified Abel polynomials, and the third one involves Hermite polynomials. The former and the latter have extensive Boltzmann-Gibbs whereas the second one (Abel) has extensive Renyi entropy. A probabilistic model is presented for this exceptional case.