# Second order possibility measure induced by a fuzzy random variable

• Inés Couso
• Susana Montes
• Pedro Gil
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 87)

## Abstract

Random sets and fuzzy random variables are commonly used to model situations where two different types of uncertainty (imprecision/vagueness and randomness) appear simultaneously. In this context, the meaning of random sets is clear. The same does not happen for the case of fuzzy random variables. The meaning depends on the particular interpretation of fuzzy sets chosen.

In this paper, we consider the possibilistic interpretation introduced by Zadeh (1978) and show some situations where the imprecise information about some characteristic of the individuals of the population may be represented by a fuzzy random variable. We deal with the concept of induced probability measure in this more general context. We examine different ways to extend this definition to the case of random sets, and show the advantages and disadvantages of each one. As a generalization of these studies, we propose a new way to describe the available information about the “original” probability measure when fuzzy random variables are used. The model introduced is closely related to second order possibility measures, recently studied by several authors.

## Keywords

Probability Measure Multivalued Mapping Polish Space Fuzzy Random Variable Initial Space

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