Entropy Optimization, Maxwell–Boltzmann, and Rayleigh Distributions

Abstract

In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations.

Keywords

• complex Maxwell–Boltzmann and Rayleigh densities
• multivariate and matrix-variate densities
• matrix-variate pathway models
• type-1, type-2 beta densities
• generalized gamma
• generalized entropy