A Remainder Term Estimate for the Normal Approximation in Classical Occupancy

Abstract

Let balls be thrown successively at random into $N$ boxes, such that each ball falls into any box with the same probability $1/N$. Let $Z_n$ be the number of occupied boxes (i.e., boxes containing at least one ball) after $n$ throws. It is well known that $Z_n$ is approximately normally distributed under general conditions. We give a remainder term estimate, which is of the correct order of magnitude. In fact we prove that $0.087/\max(3, DZ_n) \leqq \sup_x |P(Z_n < x) - \Phi((x - EZ_n)/DZ_n)| \leqq 10.4/DZ_n.$