Polya Urn Models

Many problems in probability can be represented as a model of drawing and replacing colored balls from urns according to some rule. A Polya urn model has balls of k possible colors. A single ball is sampled uniformly from the urn at each discrete time step. Depending on the color sampled, balls of various colors are added to or removed from the urn. We will examine asymptotic results regarding the distribution of balls over time and how embedding this model into a continuous setting can obtain further results. We will also present applications of the Polya model to problems in biology and computer science