Revisiting the Classical Occupancy Problem

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 classi...

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Bibliographic Details
Published in:The American statistician 2009-11, Vol.63 (4), p.356-360
Main Authors: Williamson, Patricia Pepple, Mays, Darcy P., Abay Asmerom, Ghidewon, Yang, Yingying
Format: Article
Language:English
Subjects:
Approximation
Birthdays
Boggarts
Counting rules
Distribution theory
Exact sciences and technology
General topics
Mathematical analysis
Mathematical induction
Mathematics
Missiles
Pascal's triangle
Probabilities
Probability and statistics
Probability distribution
Probability distributions
Probability theory and stochastic processes
Random variable
Random variables
Sciences and techniques of general use
Statistical variance
Statistics
Stirling numbers of the second kind
Teacher's Corner
Variance analysis
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Summary: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.
ISSN:0003-1305
1537-2731
DOI:10.1198/tast.2009.08104