The observed distribution of non-zero rainfall totals for a given month is often modelled using a maximum likelihood estimate for the Gamma distribution. In this paper we show that a Gamma distribution can be regarded as a zero order approximation to a density distribution constructed by a series of associated Laguerre polynomials. The coefficients of the series are easily calculated and used to improve the shape of the initial approximation by adjusting higher order moments. We show that this more general method models joint probability distributions for two or more months and in particular that the series model does not require an assumption of independence between months. Finally we explore how the series method generates simulated data that is statistically indistinguishable from the observed data. We illustrate our methods on a case study at Mawson Lakes and although monthly correlations may not be significant we note that the rainfall records at Koonamore Station show many significant correlations for successive months.
