Three-Parameter Probability Distributions

The lognormal, Weibull, Pearson type 3, and log Pearson type 3, each a three-parameter distribution, were evaluated in a generalized fashion in terms of the dimensionless variate K (K=X/υ_x, in which X equals randon variable, and υ_n equals its mean) which has a population mean of unity. The bounds of the distributions, areas of the portions of distributions in the negative region of variate when they enter such regions, and the differences in some important quantiles among the four distributions, are presented. The four distributions become less applicable for hydrologic frequency analysis as they deviate more and more from their two-parameter counterparts (lognormal in the case of log Pearson). When they have well-applicable properties, their quantile values differ little for some or all distributions indicating that choice of a distribution makes little difference. Some guidelines are provided for selecting the best applicable distribution for a given hydrologic sample.