Research paper
Normalization of a hydrologic sample probability density function by transform optimization

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Abstract

Transformation of variates is the conventional procedure for deriving a probability density function of a variate y when a probability density 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 known and the transform is treated as a differential equation which is solved to yield an optimal variate transform function x = f(y). Specifically, 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 the normal distribution can then be used for estimating confidence intervals of the mean and tolerance limits of outer values. Such estimates of risk and uncertainty, when de-transformed back to the sampled variate of interest, x, are valuable tools in hydrology and water resource and environmental protection analysis and planning. The numerical methodology for solving the differential equation is not specific to the particular problem, and can be extended to other situations and other probability density functions.

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