Some Comments Upon Entropy Maximizing

Abstract

It is interesting to note that the absolutely continuous distributions discussed in the previous section are the solutions to the following three problems:

• max H subject to $\left( {\begin{array}{*{20}{c}} {EX = \mu ,} \\ {Var{\text{ }}X = {{\sigma }^{2}},} \\ \end{array} } \right.$, normal distribution

• max H subject to EX = 1/μ, negative exponential distribution

• max H subject to $\begin{array}{*{20}{c}} {\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {EX = \nu /\alpha ,} \\ {{\text{E log X = - c - log }}\alpha + \sum\limits_{{k = 1}}^{{\nu - 1}} {\frac{1}{k},} } \\ \end{array} {\text{ }}} \\ \end{array} } \right.} \\ \end{array}$,

• where C is Euler’s constant; gamma distribution

(Kagan-Linnik-Radhakrishna Rao, 1973).