## The Annals of Mathematical Statistics

### A Note on the Classical Occupancy Problem

C. J. Park

#### 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 $S_0$.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 5 (1972), 1698-1701.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692405

Digital Object Identifier
doi:10.1214/aoms/1177692405

Mathematical Reviews number (MathSciNet)
MR346870

Zentralblatt MATH identifier
0247.60020

JSTOR