Probability density functions of fuzzy random variables
The notion of fuzzy random variable was introduced by \textit{H. Kwakernaak} [Inf. Sci. 15, 1-29 (1978; Zbl 0438.60004)] and \textit{M. L. Puri} and \textit{D. A. Ralescu} [J. Math. Anal. Appl. 114, 409-422 (1986; Zbl 0592.60004)]. The aim of the present paper is to propose a new concept of a fuzzy random variable and to construct the (fuzzy) probability density function (PDF) of a fuzzy random variable from the known PDF. A basic mathematical result is L. A. Zadeh's ``resolution identity'' which provides a closed fuzzy number from a family of closed intervals. Using the same technique, the author constructs the (fuzzy) PDF of a fuzzy random variable from the known PDF. In order to find the membership of (fuzzy) PDF of fuzzy observation from the fuzzy random variable, the original problem is transformed into a nonlinear programming problem. A computational technique is proposed to evaluate the membership of probability density of fuzzy observation from a fuzzy random variable. The numerical method is illustrated by finding the PDF membership of gamma distribution.
scientific article; zbMATH DE number 1339612