On Additive and Non-Additive Entropies of Continuous Random Variables

This paper investigates the maximization of the Havrda-Charvat α-entropy for continuous random variables under various constraints. A general form of the probability density function that maximizes this entropy subject to a set of moment constraints is derived. This solution is then applied to three fundamental cases. For a random variable on a finite interval, the entropy is maximized by the uniform distribution. For a non-negative random variable with a specified mean, the maximizing distribution is shown to be a Pareto Type II (Lomax) distribution, valid for the entropy order 1/2 < α < 1. Finally, for a random variable with a specified second moment, the maximum entropy distribution is a scaled Student's t-distribution, which holds for 1/3 < α < 1. The explicit forms of the maximizing distributions and the corresponding maximum entropy values are presented for each scenario, extending the principle of maximum entropy to this generalized framework.