NORMALIZATION OF A HYDROLOGIC SAMPLE PROBABILITY DENSITY-FUNCTION BY TRANSFORM OPTIMIZATION

Transformation of variates is the conventional procedure for deriving a probability density function of a variate y when a probability densi ty function of a variate x and a function y = f(x) are known. In this study the probability density functions p(x) and p(y) are assumed know n and the transform is treated as a differential equation which is sol ved to yield an optimal variate transform function x = f(y). Specifica lly, p(x) is considered a sample probability density function and p(y) is the normal probability density function. The solution for x = f(y) thus provides an optimal normalization of the sample. Properties of t he normal distribution can then be used for estimating confidence inte rvals of the mean and tolerance limits of outer values. Such estimates of risk and uncertainty, when de-transformed back to the sampled vari ate of interest, x, are valuable tools in hydrology and water resource and environmental protection analysis and planning. The numerical met hodology for solving the differential equation is not specific to the particular problem, and can be extended to other situations and other probability density functions.