On discrete distributions generated from drawing balls with bivariate labels

ABSTRACT

In this article, we consider urn models under three types of sampling schemes in terms of the probability-generating functions. The tools are developed for the evaluation of the distributions arising from the urn models along with some examples. Furthermore, the distributions are investigated by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of urn models. As examples, we propose new class of probability models, which are called multiple-player problems and examine their properties. Finally, we treat the parameter estimation problem in the waiting time distributions with a numerical example.

KEYWORDS:

• Bell polynomials
• Birthday problem
• Bivariate distribution
• Coupon collection problem
• Double-generating function
• Multiple-player problem
• Pólya urn model
• Probability function
• Probability-generating function
• Sooner and later waiting times
• Urn model

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62E15
• Secondary 60E05

Acknowledgments

The authors wish to thank the editor and the referees for careful reading of our paper and helpful suggestions which led to improved results.