### Fitting theoretical probability distributions

Probability distributions which are most commonly used for modelling annual maxima series are:

- Lognormal distribution
- Pearson type 3 distribution
- Log-Pearson type 3 distribution
- Gumbel distribution
- Generalized extreme value distribution

See Probability distributions for hydrologic variables for description of the above distributions.

In practice, several distributions are fitted to sample data. The final choice of the distribution is made according to the criteria described in Principles for selecting a distribution.

**Estimation of parameters of theoretical distributions. **Fitting a theoretical probability distribution to sample data is accomplished by estimating the parameters of the distribution on the basis of observed data. There are several methods for estimation of parameters. The most straightforward and the most commonly used method is the *method of moments*.

The principle of the method of moments is that the moments of a theoretical distribution are equal to the corresponding sample moments. In this way, by setting the first *r* distribution moments equal to the first *r* sample moments, a set of *r* equations is obtained with *r* unknown distribution parameters. The solution of these equations are then the estimates of distribution parameters. For example, for a two-parameter distribution, first two distribution moments and sample mean and standard deviation are used to estimate the two parameters.

The above distributions and their parameter estimates are described in Quick reference to distributions used in flood frequency analysis (PDF).

**Calculation of theoretical distributions. **Once the parameters of the distribution are estimated, calculations can be made in two directions:

- direct calculation: calculate probability for a given value
*x*, or - inverse calculation: calculate the value
*x* for* *a given probability.

The basis for these calculations is the cumulative distribution function *F*(*x*), representing the probability of nonexceedance *P*{*X* £ *x*}. If the flood probability is expressed in a different way (as the probability of exceedance or as the return period), it should be "converted" into probability of nonexceedance.

On how to calculate direct and inverse c.d.f. for selected theoretical probability distributions see Quick reference to distributions used in flood frequency analysis (PDF).