Revisiting the Classical Occupancy Problem

Patricia Pepple Williamson Patricia Pepple Williamson is Instructor, Department of Statistics, Case Western Reserve University, Cleveland, OH 44106-7054 . Darcy P. Mays is Associate Professor and Chair, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083 . Ghidewon Abay Asmerom is Associate Professor and Assistant Chair, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014 . Yingying Yang is Graduate, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083. The authors thank Wojbor Woyczynski, Case Western Reserve University, for his helpful comments. The authors also thank the referee, associate editor, and editor for the very thorough and helpful review that contributed to substantial improvement of the article.

Darcy P. Mays Patricia Pepple Williamson is Instructor, Department of Statistics, Case Western Reserve University, Cleveland, OH 44106-7054 . Darcy P. Mays is Associate Professor and Chair, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083 . Ghidewon Abay Asmerom is Associate Professor and Assistant Chair, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014 . Yingying Yang is Graduate, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083. The authors thank Wojbor Woyczynski, Case Western Reserve University, for his helpful comments. The authors also thank the referee, associate editor, and editor for the very thorough and helpful review that contributed to substantial improvement of the article.

Ghidewon Abay Asmerom Patricia Pepple Williamson is Instructor, Department of Statistics, Case Western Reserve University, Cleveland, OH 44106-7054 . Darcy P. Mays is Associate Professor and Chair, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083 . Ghidewon Abay Asmerom is Associate Professor and Assistant Chair, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014 . Yingying Yang is Graduate, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083. The authors thank Wojbor Woyczynski, Case Western Reserve University, for his helpful comments. The authors also thank the referee, associate editor, and editor for the very thorough and helpful review that contributed to substantial improvement of the article.

Yingying Yang Patricia Pepple Williamson is Instructor, Department of Statistics, Case Western Reserve University, Cleveland, OH 44106-7054 . Darcy P. Mays is Associate Professor and Chair, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083 . Ghidewon Abay Asmerom is Associate Professor and Assistant Chair, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014 . Yingying Yang is Graduate, Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284-3083. The authors thank Wojbor Woyczynski, Case Western Reserve University, for his helpful comments. The authors also thank the referee, associate editor, and editor for the very thorough and helpful review that contributed to substantial improvement of the article.

Abstract

This article is offered as an example of a probability problem, simple to state, but rich enough to permit a variety of explorations by students with adequate analytical skills in an introductory course in probability and statistics. It involves a discrete random variable originating from the classical occupancy problem. This random variable X is defined to be how many of N elements are selected by or assigned to K individuals when each of the N elements is equally likely to be chosen by or assigned to any of the K individuals. The probability distribution of this random variable is derived utilizing various counting rules, properties of probability, and observing trends. The distribution involves the Stirling numbers of the second kind and some asymptotic results are given. Several interesting applications of this random variable, including a variant of the birthday problem, are described.

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