A distribution function for particle populations having a finite size range and a mode independent of the spread

Abstract

A distribution function ▪and its cumulative form ▪ are described and their mathematical properties analyzed. Since f(0) = f(a) = 0, this function can be applied to real particle populations, having a definite size range 0 < x < a determined by physical considerations. Unlike most asymmetric distribution functions, the mode estimate μ can be varied independently of the spread measure C, thus being more consistent with real life situations where a process that affects one need not affect the other in a predetermined manner. The mathematical structure of f(x) allows for both symmetry around the mean and skewness to either right or left, depending on whether μ = a 2 , μ > a 2 or μ < a 2 , respectively. This permits the simulation of extreme changes in particle size distribution patterns without changing the model format. The parameter B is an arbitrary constant whose effect on the distribution curve shape is of little significance as long as its magnitude remains small relative to that of a. Although the integrals that appear in f(x) and F(x) cannot be easily solved analytically, their values can conveniently be computed by standard numerical methods employing a microcomputer.