Review of pólya Urn models by Hosam Mahmoud

Consider an urn containing a number of colored balls. Suppose that you are assigned the task of repeatedly reaching blindly into it, picking out a ball, replacing it, and then depending on the color of that ball adding a specified number of new colored balls into the urn. Such an urn and associated sampling-replacement scheme is referred to as a “Pólya urn,” and given this description a number of mathematical questions can be asked: What is the long-term proportion of balls expected to be allocated to one color? After a given number of samples are drawn, what is the distribution of the number of balls of a particular color? How do the answers to these questions change as the rules for the urn are generalized for example, what if conditional on drawing a ball of a particular color a random rather than specified set of balls of various colors are readded to the urn, or what if rather than drawing only one ball at a time, multiple balls are drawn? More applied researchers might utilize these mathematical results to attempt to model a variety of physical systems, and in so doing create new demands for both further generalizations of the Pólya Urn and mathematical technique for analyzing the behavior of these urns. In Pólya Urn Models (2009), Hosam Mahmoud provides both an overview of the mathematical tools used to study Pólya urns and examples of their application to multiple problems in computer science and the biosciences. The primary requirement for the book is a familiarity with probability and stochastic processes, although additional background in differential equations and combinatorics is useful. There is substantial