A Note on the Classical Occupancy Problem

Abstract

Assume that $n$ balls are randomly distributed into $N$ equiprobable cells. The ball is presumed to have probability $p, 0 < p < 1$ of staying in the cell and $(1 - p)$ of falling through. Let $S_0$ denote the number of empty cells. In this note we establish the asymptotic normality of $S_0$ as $n$ and $N$ tend to infinity so that $np/N \rightarrow c > 0, np/N^{\frac{5}{6}} \rightarrow \infty$ and $n/N \rightarrow 0$, or $3np/N - \log N \rightarrow - \infty$ and $n/N \rightarrow \infty$. We accomplish this by estimating the factorial cumulants of $S_0$.