The index of coincidence for the binomial distribution is log-convex

TL;DR

This paper proves that the sum of squared probabilities of a binomial distribution is a log-convex function of its parameter, confirming a conjecture from 2014, with applications to entropy measures.

Contribution

It provides a proof of the log-convexity of the sum of squared binomial probabilities, completing a conjecture from 2014 and exploring related entropy applications.

Findings

Sum of squared binomial probabilities is log-convex in x

Confirms a conjecture from 2014

Applications to Rényi and Tsallis entropies

Abstract

We consider the binomial distribution with parameters $n$ and $x$, and show that the sum of the squared probabilities is a log-convex function of $x$. This completes the proof of a conjecture formulated in 2014. Applications to R\'{e}nyi and Tsallis entropies are given.