Three Distributions in the Extended Occupancy Problem

The classical and extended occupancy distributions are useful for examining the number of occupied bins in problems involving random allocation of balls to bins. We examine the extended occupancy problem by framing it as a Markov chain and deriving the spectral decomposition of the transition probab...

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Bibliographic Details
Published inMethodology and computing in applied probability Vol. 25; no. 4; p. 84
Main Author O’Neill, Ben
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2023
Springer Nature B.V
Subjects
Bins
Business and Management
Combinatorial analysis
Economics
Electrical Engineering
Life Sciences
Markov analysis
Markov chains
Mathematics and Statistics
Probability
Random variables
Statistics
Stochastic models
Transition probabilities
Markov chain
Spillage distribution
Pure birth process
Occupancy distribution
Extended occupancy problem
Negative-occupancy distribution
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Summary:The classical and extended occupancy distributions are useful for examining the number of occupied bins in problems involving random allocation of balls to bins. We examine the extended occupancy problem by framing it as a Markov chain and deriving the spectral decomposition of the transition probability matrix. We look at three distributions of interest that arise from the problem, all involving the noncentral Stirling numbers of the second kind. These distributions give a useful generalisation to the binomial and negative-binomial distributions. We examine how these distributions relate to one another, and we derive recursive properties and mixture properties that characterise the distributions.
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ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-023-10053-y