Multimodality of the Markov binomial distribution

TL;DR

This paper analyzes the shape of the Markov binomial distribution's probability mass function, providing conditions for unimodality, bimodality, and trimodality, along with formulas for mean and variance, relevant for engineering models.

Contribution

It offers necessary and sufficient conditions for the distribution's shape and derives explicit formulas for mean and variance, enhancing understanding of its properties.

Findings

Conditions for unimodal, bimodal, and trimodal PMFs

Closed-form expression for variance

Conditional mean and variance formulas

Abstract

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal or trimodal. These are useful to analyze the double-peaking results from a PDE reactive transport model from the engineering literature. Moreover, we give a closed form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time $n$.