  We discuss various statistical distributions of earthquake numbers.
Previously we derived several discrete distributions to describe earthquake
numbers for the branching model of earthquake occurrence: these distributions
are the Poisson, geometric, logarithmic, and the negative binomial (NBD). The
theoretical model is the `birth and immigration' population process. The first
three distributions above can be considered special cases of the NBD. In
particular, a point branching process along the magnitude (or log seismic
moment) axis with independent events (immigrants) explains the
magnitude/moment-frequency relation and the NBD of earthquake counts in large
time/space windows, as well as the dependence of the NBD parameters on the
magnitude threshold (magnitude of an earthquake catalog completeness). We
discuss applying these distributions, especially the NBD, to approximate event
numbers in earthquake catalogs. There are many different representations of the
NBD. Most can be traced either to the Pascal distribution or to the mixture of
the Poisson distribution with the gamma law. We discuss advantages and
drawbacks of both representations for statistical analysis of earthquake
catalogs. We also consider applying the NBD to earthquake forecasts and
describe the limits of the application for the given equations. In contrast to
the one-parameter Poisson distribution so widely used to describe earthquake
occurrence, the NBD has two parameters. The second parameter can be used to
characterize clustering or over-dispersion of a process. We determine the
parameter values and their uncertainties for several local and global catalogs,
and their subdivisions in various time intervals, magnitude thresholds, spatial
windows, and tectonic categories.
