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

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.