The entropyHof an absolutely continuous distribution with probability density functionp(x)is defined asH = - \int p(x) \log p(x) dx. The formal maximization ofH, subject to the moment constraints\int x^r p(x) dx = \mu_r, r = 0,1,\cdots,m, leads top(x) = \exp (- \sum_{r=0}^m \lamnbda_r x^r), where the\lambda_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,\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_1^2is discussed in some detail.

The problem of estimating a probability density functionp(x)over[0, \infty)when several low-order moments are known is considered. A computational procedure, based on information theory, is used to obtain the maximum entropy estimates ofp(x)in a number of cases. The situation when the first two moments only are known is considered in some detail. A table is included for estimatingp(x)when given\mu_1 ^{\prime} , \mu_2 ^{\prime}with\mu_2 ^{\prime} \leq 2(\mu_1 ^{\prime}) ^ 2.

Numerical methods are described for finding the boundary separating the continuation and stopping regions for a class of sequential decision problems whose minimal cost function satisfies a parabolic partial differential equation. The problem is shown to be similar to that of finding the moving boundary in Stefan problems, and techniques used in the solution of Stefan problems are modified for use in decision problems. Numerical results for a particular sequential decision problem are given.

Lanczos τ-methods are used to solve the ordinary differential equations with two-point boundary conditions which result from a finite-difference approximation to a Stefan problem. This technique, combined with the use of canonical polynomials, provides the basis of a computer program to solve the Stefan problem which, under certain conditions, is more efficient than the Douglas–Gallie method. Numerical results from the program are given.