On Some Properties of the Beta Normal Distribution

TL;DR

This paper explores the mathematical properties, bimodality, hazard rate, moments, and entropy of the beta normal distribution, enhancing understanding of its behavior and potential applications.

Contribution

It provides a comprehensive analytical study of the beta normal distribution, including explicit formulas and insights into its bimodality and limiting behaviors.

Findings

Derived explicit expressions for moments and entropy.

Analyzed the bimodality and hazard rate function.

Examined the distribution's limiting behavior.

Abstract

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.