An examination of the spillage distribution

TL;DR

This paper analyzes a specific probability distribution related to the spillage number in an extended balls-in-bins model, deriving generating functions, moments, recursive algorithms, and asymptotic behavior.

Contribution

It introduces a novel distribution linked to the extended balls-in-bins model and provides methods for its computation and analysis.

Findings

Derived generating functions for the distribution

Developed a recursive algorithm for computing distribution values

Analyzed the asymptotic behavior and approximation performance

Abstract

We examine a family of discrete probability distributions that describes the "spillage number" in the extended balls-in-bins model. The spillage number is defined as the number of balls that occupy their bins minus the total number of occupied bins. This probability distribution can be characterised as a normed version of the expansion of the noncentral Stirling numbers of the second kind in terms of the central Stirling numbers of the second kind. Alternatively it can be derived in a natural way from the extended balls-in-bins model. We derive the generating functions for this distribution and important moments of the distribution. We also derive an algorithm for recursive computation of the mass values for the distribution. Finally, we examine the asymptotic behaviour of the spillage distribution and the performance of an approximation to the distribution.