Maximum-entropy distributions having prescribed first and second moments (Corresp.)

The entropyHof an absolutely continuous distribution with probability density functionp(x)is defined asH = - ∫ p(x) log p(x) dx. The formal maximization ofH, subject to the moment constraints∫ x^r p(x) dx = μ_r, r = 0,1,⋯,m, leads top(x) = exp (- ∑ᵣ₌₀^m _r x^r), where theλ_rhave to be chosen so as to satisfy the moment constraints. Only the casem = 2is considered. It is shown that whenxhas finite range, a distribution maximizing the entropy exists and is unique. When the range is[0,∞), the maximum-entropy distribution exists if, and only if,μ_2 ≤ 2 μ_1^2, and a table is given which enables the maximum-entropy distribution to be computed. The caseμ_2 > 2 μ_1^2is discussed in some detail.