The Total Variation Distance Between the Binomial and Poisson Distributions

Abstract

The exact total variation distances are obtained between a binomial distribution with parameters $n$ and $p$ and Poisson distributions with means $np$ and $-n \log(1 - p)$, for small values of $p$. It is shown that the latter distance is smaller for $0 < p < c_n$ and larger for $c_n < p < a'_{n0}$, where as $n \rightarrow \infty, nc_n \rightarrow 1.596 \ldots$ and $na'_{n0} \rightarrow 3.414 \ldots.$