Orthogonal polynomials in the approximation of probability distributions.

Abstract

An approach to the problem of approximating a continuous probability distribution with a series in orthogonal polynomials is presented. The approach is first motivated with a discussion of theoretical distributions which are inherently difficult to evaluate. Additionally, a practical application which involves such a distribution is developed. The three families of orthogonal polynomials that pertain to the methodology--the Hermite, Laguerre, and Jacobi--are then introduced. Important properties and characterizations of these polynomials are given to lay the mathematical framework for the orthogonal polynomial series representation of the probability density function of a continuous random variable. This representation leads to a similar series for the cumulative distribution function, which is of more practical use for computing probabilities associated with the random variable. It is demonstrated that the representations require only the moments and the domain of the random variable to be known. Relationships of the Hermite, Laguerre, and Jacobi series approximations to the normal, gamma, and beta probability distributions, respectively, are also formally established. Examples and applications of the series are given with appropriate analyses to validate the accuracy of the approximation.