*J. Japan Statist. Soc.,* Vol. 31 (No. 2), pp. 193-205, 2001

## Pólya Urn Models Under General Replacement Schemes

Kiyoshi Inoue and Sigeo Aki

**Abstract. **
In this paper, we consider a Pólya urn model containing balls of *m* different labels under a general replacement scheme, which is characterized by an *m* *m* addition matrix of integers without constraints on the values of these *m*^{2} integers other than non-negativity. This urn model includes some important urn models treated before. By a method based on the probability generating functions, we consider the exact joint distribution of the numbers of balls with particular labels which are drawn within *n* draws. As a special case, for *m* = 2, the univariate distribution, the probability generating function and the expected value are derived exactly. We present methods for obtaining the probability generating functions and the expected values for all *n* exactly, which are very simple and suitable for computation by computer algebra systems. The results presented here develop a general workable framework for the study of Pólya urn models and attract our attention to the importance of the exact analysis. Our attempts are very useful for understanding non-classical urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results.

*Key words and phrases*:
Pólya urn, replacement scheme, addition matrix, probability generating functions, double generating functions, expected value.

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