  The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter
generalisation of the Poisson distribution which has received some attention in
the statistics literature in recent years by offering flexible generalisations
of some well-known models. In this work, we begin by establishing some
properties of both the CMP distribution and an analogous generalisation of the
binomial distribution, which we refer to as the CMB distribution. We also
consider some convergence results and approximations, including a bound on the
total variation distance between a CMB distribution and the corresponding CMP
limit.
