Polya Urn Models  book cover
1st Edition

Polya Urn Models

ISBN 9781420059830
Published June 30, 2008 by Chapman and Hall/CRC
312 Pages 11 B/W Illustrations

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Book Description

Incorporating a collection of recent results, Pólya Urn Models deals with discrete probability through the modern and evolving urn theory and its numerous applications.

The book first substantiates the realization of distributions with urn arguments and introduces several modern tools, including exchangeability and stochastic processes via urns. It reviews classical probability problems and presents dichromatic Pólya urns as a basic discrete structure growing in discrete time. The author then embeds the discrete Pólya urn scheme in Poisson processes to achieve an equivalent view in continuous time, provides heuristical arguments to connect the Pólya process to the discrete urn scheme, and explores extensions and generalizations. He also discusses how functional equations for moment generating functions can be obtained and solved. The final chapters cover applications of urns to computer science and bioscience.

Examining how urns can help conceptualize discrete probability principles, this book provides information pertinent to the modeling of dynamically evolving systems where particles come and go according to governing rules.

Table of Contents

Urn Models and Rudiments of Discrete Probability
Probability Axioms
Random Variables
Independence of Discrete Random Variables
Realization of Basic Discrete Distributions via Urns
The Poisson Process
Markov Chains
Some Classical Urn Problems
Ballot Problems
Occupancy Problems
Coupon Collector’s Problems
The Gambler’s Ruin
Banach’s Matchbox Problem
Pólya Urn Models
The Pólya–Eggenberger Urn
Bernard Friedman’s Urn
The Bagchi–Pal Urn
The Ehrenfest Urn
The Pólya Process
Poissonization of Multicolor Deterministic Schemes
The Deterministic Monochromatic Pólya Process
The Pólya–Eggenberger Process
The Ehrenfest Process
Poissonization of Schemes with Random Entries
The Depoissonization Heuristic
Depoissonization of Dichromatic Pólya Processes
Depoissonization of Invertible Processes
Depoissonization Examples
Urn Schemes with Random Replacement
Extended Urn Schemes
Deterministic Dichromatic Extended Urn Schemes
Dichromatic Extended Urn Schemes with Random Entries
Extended Multicolor Urn Schemes
Analytic Urns
Enumerating Histories in Balanced Schemes
Analytic Urn Examples
Applications of Pólya Urns in Informatics
Search Trees
Recursive Trees
Urn Schemes in Bioscience
Evolution of Species
Competitive Exclusion
Epidemiology: Kriz’ Polytonomous Urn Scheme
Clinical Trials
Urns Evolving by Multiple Drawing
Urns Evolving by Drawing Pairs of Balls
The Chen–Wei Urn
Linear Rectangular Urns
Application of Rectangular Schemes in Random Circuits
Answers to Exercises
Bibliographic Notes
Exercises appear at the end of each chapter.

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"Pólya Urn Models describes the established theory and details numerous new developments since the seminal 1977 text Urn Models and Their Application: An Approach to Modern Discrete Probability Theory by Johnson and Kotz. … the second half of Mahmoud’s book largely covers new advances since Johnson and Kotz’s book … This latter material makes the current text unique. … The end-of-chapter exercises (of greatly varying difficulty) have excellent in-depth solutions at the end of the text. … The book is exceptionally well written, concise, and compelling. The author obviously commands this material, and his enthusiasm is unmistakable."
—Matthew Bognar, Journal of the American Statistical Association, September 2014, Vol. 109