Three Distributions in the Extended Occupancy Problem

TL;DR

This paper analyzes the extended occupancy problem by modeling it as a Markov chain, deriving spectral decompositions, and exploring three related distributions involving noncentral Stirling numbers, generalizing binomial and negative-binomial distributions.

Contribution

It introduces a Markov chain framework for the extended occupancy problem and derives spectral and recursive properties of three new distributions involving noncentral Stirling numbers.

Findings

Derived spectral decomposition of the transition matrix.

Established recursive and mixture properties of the distributions.

Showed how these distributions generalize binomial and negative-binomial distributions.

Abstract

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.