商品簡介
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.
目次
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 Exchangeability Some Classical Urn Problems Ballot Problems Occupancy Problems Coupon Collector’s Problems The Gambler’s Ruin Banach’s Matchbox Problem Pólya Urn Models Tenability The Pólya–Eggenberger Urn Bernard Friedman’s Urn The Bagchi–Pal Urn The Ehrenfest Urn Poissonization 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 Martingales 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 Notation Bibliographic Notes Bibliography Index Exercises appear at the end of each chapter.
