Finite Fourier Probability Distribution and Applications

A probability distribution is developed to describe data collected from processes that are diffusion driven, in addition to data sets in which the range of the random variable has a fixed lower bound, a fixed upper bound, or both. Chemical equilibrium relationships constrain some data such as water hardness to a fixed lower limit, while chemical solubility relationships establish a fixed upper bound to other water quality data. The solution of a 1D diffusion equation subject to an impulse loading of mass can be adapted as a probability distribution. In this study, the focus is on the solution of the diffusion equation using the integral transform technique when the solution range is 0 ⩽ x ⩽ L. The solution is adapted as a probability distribution function that is referred to as the finite Fourier distribution. The finite Fourier distribution and the three-parameter gamma distributions were applied to water quality data from the Mississippi River for hardness and for sulfate and magnesium concentrations. The Kolmogorov-Smirnov one-sample goodness-of-fit test was applied to compare the two distributions for each of the data sets. Both finite Fourier and three-parameter gamma distributions were accepted. However, three-parameter gamma distributions did not model small concentrations as well as did the finite Fourier distribution. The advantages of the finite Fourier distribution are that the new distribution is derived from a physical process related to the data collected and that the parameters provide a clearer picture of the shape of the distribution.