Constructing membership functions using statistical data

Determination of the membership functions is vital in practical applications of the fuzzy set theory. This paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function (pdf) in the universe of discourse. The method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf. It is show that, for any pdf, the method is capable of generating membership functions in accordance with the possibility-probability consistency principle. Membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented.