A Possibilistic Interpretation of the Expectation of a Fuzzy Random Variable

Abstract

Since their first introduction, fuzzy random variables have been given a number of different definitions. In [10], Kratschmer gives an unified approach of all of them. All of these authors try to model situations where both randomness and imprecision are present, and they define a fuzzy random variable as a map assigning to any element of the initial space a fuzzy subset of the final space; however, they differ in the measurability condition imposed on this map and in the characteristics of the final space. On the other hand, the study of statistical parameters, such as expectation and variance, of a fuzzy random variable has followed two different approaches: some authors define them as fuzzy sets ([1, 11, 14, 17]); others as (crisp) numerical values, as in [9, 12, 13]. One of the reasons for the existence of these different approaches is that a fuzzy set can be given many different interpretations, as the survey conducted in [7] testifies; and any of these interpretations can be carried over to fuzzy random variables and their parameters. In the present paper, we intend to give fuzzy random variables a possibilistic interpretation. The value of a fuzzy set in a point will represent a degree of possibility, which is a specific type of upper probability. We shall see that this interpretation fits nicely into the framework of the theory of imprecise probabilities ([18]), and we shall be able to associate with any statistical parameter of interest an interval of possible values. This is a compromise between the two approaches considered above (precise numerical values and fuzzy sets): as sets of possible values these intervals have a straightforward interpretation in the context of the theory of imprecise probabilities, and the fact that these intervals do not 134 Couso, Miranda and de Cooman generally reduce to a precise single value allows us to take into account the imprecision that a fuzzy random variable represents.