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

Author

Wragg, A.

Volume

Issue

fYear

1973

fDate

9/1/1973 12:00:00 AM

Firstpage

689

Lastpage

693

Abstract

The entropy $H$ of an absolutely continuous distribution with probability density function $p(x)$ is defined as $H = - \\int p(x) \\log p(x) dx$ . The formal maximization of $H$ , subject to the moment constraints $\\int x^r p(x) dx = \\mu_r, r = 0,1,\\cdots ,m$ , leads to $p(x) = \\exp (- \\sum _{r=0}^m lamnbda_r x^r)$ , where the $\\lambda _r$ have to be chosen so as to satisfy the moment constraints. Only the case $m = 2$ is considered. It is shown that when $x$ has finite range, a distribution maximizing the entropy exists and is unique. When the range is $[0,\\infty )$ , the maximum-entropy distribution exists if, and only if, $\\mu_2 \\leq 2 \\mu_1^2$ , and a table is given which enables the maximum-entropy distribution to be computed. The case $\\mu_2 > 2 \\mu_1^2$ is discussed in some detail.

Keywords

Entropy functions; Probability functions; Density functional theory; Distributed computing; Entropy; Equations; Gaussian distribution; Lagrangian functions; Mathematics; Measurement uncertainty; Probability density function;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1973.1055060

Filename

1055060

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=919996