  We explore negative dependence and stochastic orderings, showing that if an
integer-valued random variable $W$ satisfies a certain negative dependence
assumption, then $W$ is smaller (in the convex sense) than a Poisson variable
of equal mean. Such $W$ include those which may be written as a sum of totally
negatively dependent indicators. This is generalised to other stochastic
orderings. Applications include entropy bounds, Poisson approximation and
concentration. The proof uses thinning and size-biasing. We also show how these
give a different Poisson approximation result, which is applied to mixed
Poisson distributions. Analogous results for the binomial distribution are also
presented.
