Fuzzy Set-Valued Random Variables

Abstract

In practice we are often faced with random experiments whose outcomes are not numbers but are expressed in inexact linguistic terms. For example, consider a group of individuals chosen at random who are questioned about the weather in a particular city on a particular winter’s day. The resulting data of this random experiment would be linguistic terms such as ‘cold’, ‘more or less cold’, ‘very cold’, ‘extremely cold, which can be described by fuzzy sets, introduced by Zadeh in 1965 in his paper [234], rather than by a single real number or subsets of real numbers. A natural question which arises with reference to this example is: what is the average opinion about the weather in that particular city on a particular day? A possible way of handling ‘data’ like this is by using the concepts of fuzzy sets and expectations of fuzzy set-valued random variables. Fuzzy set-valued random variables are random variables whose values are not numbers or sets but fuzzy sets.