Notes on Feige's gumball machines problem

TL;DR

This paper provides a detailed proof of Feige's conjecture regarding the maximum probability that the sum of n independent, identically distributed non-negative integer-valued variables with mean 1 exceeds n, focusing on the identically distributed case.

Contribution

It offers a rigorous proof of Feige's conjecture in the i.i.d. case and reduces the general case to two-point distributions, advancing understanding of sum probabilities.

Findings

Confirmed Feige's conjecture for i.i.d. variables

Reduced the general case to two-point distributions

Provided detailed proof structure for the conjecture

Abstract

We give a detailed proof, in the identically distributed case, of a conjecture of Feige about the maximum probability that the sum of n independent non-negative integer valued random variables, each of mean 1, exceeds n. The general case is reduced to two-point distributions.