Introduction to Stochastic Models

Chapter 2

Simple Stochastic Models

This chapter presents stochastic models that, despite their simplicity, have been important for the development of the probability theory and of random modeling.

2.1. Urn models

An urn model (or scheme) is a system formed by urns that contain balls of different colors, to which we associate an experiment of successive ball drawings, with or without replacement in the urns, following some precise rules. These rules could consist in either putting in additional balls or taking out balls from some urns, or in changing the color of some balls at different stages of the experiment. The fundamental rule for computing the probability of a certain result of an experiment is that the random drawing from an urn with s balls is uniformly distributed, i.e. the probability of drawing any of the balls is 1/s.

In fact it is possible, at least theoretically, to associate an urn model with any random experiment with finite or countable number of outcomes [PÒL 54]. This was almost the only methodology at the beginning of probability theory, although urn models are not the only concepts used for discrete probabilistic models. A book of L. Hogben, entitled Chance and Choice by Cardpack and Chessboard (two volumes, 1950 and 1955, Max Parrish, London) presents other possible modeling ideas. Nevertheless, compared to card games, dice games, or chess, urns have the advantage of not being associated with a precise number like 52, 6, or 64. One can imagine a 17-sided die, ...