  Frequency-magnitude distributions, and their associated uncertainties, are of
key importance in statistical seismology. When fitting these distributions, the
assumption of Gaussian residuals is invalid since event numbers are both
discrete and of unequal variance. In general, the observed number in any given
magnitude range is described by a binomial distribution which, given a large
total number of events of all magnitudes, approximates to a Poisson
distribution for a sufficiently small probability associated with that range.
In this paper, we examine four earthquake catalogues: New Zealand (Institute of
Geological and Nuclear Sciences), Southern California (Southern California
Earthquake Center), the Preliminary Determination of Epicentres and the Harvard
Centroid Moment Tensor (both held by the United States Geological Survey).
Using independent Poisson distributions to model the observations, we
demonstrate a simple way of estimating the uncertainty on the total number of
events occurring in a fixed time period.
