Orthogonal polynomials in the approximation of probability distributions.

Persistent Link:
http://hdl.handle.net/10150/185117
Title:
Orthogonal polynomials in the approximation of probability distributions.
Author:
Oakley, Steven James
Issue Date:
1990
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
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.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Operations research
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Systems and Industrial Engineering; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Higle, Julia L.

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleOrthogonal polynomials in the approximation of probability distributions.en_US
dc.creatorOakley, Steven Jamesen_US
dc.contributor.authorOakley, Steven Jamesen_US
dc.date.issued1990en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractAn 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.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectOperations researchen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineSystems and Industrial Engineeringen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorHigle, Julia L.en_US
dc.contributor.committeememberAskin, Ronald G.en_US
dc.contributor.committeememberSen, Suvrajeeten_US
dc.contributor.committeememberCushing, James M.en_US
dc.contributor.committeememberMaier, Robert S.en_US
dc.identifier.proquest9100046en_US
dc.identifier.oclc708415832en_US
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