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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Ines Couso, Didier Dubois, Susana Montes, Luciano Sanchez

# On various definitions of the variance of a fuzzy random variable

### Abstract

According to the current literature, there are two different approaches to the definition of the variance of a fuzzy random variable. In the first one, the variance is defined as a fuzzy interval, offering a gradual description of our incomplete knowledge about the variance of an underlying, imprecisely observed, classical random variable. In the second case, the variance of the fuzzy random variable is defined as a crisp number, that makes it easier to handle in further processing. In this work, we introduce yet another definition of the variance of a fuzzy random variable, in the context of the theory of imprecise probabilities. The new variance is not defined as a fuzzy or crisp number, but it is a real interval, which is a compromise between both previous definitions. Our main objectives are twofold: first, we show the interpretation of the new variance and, second, with the help of simple examples, we demonstrate the usefulness of all these definitions when applied to particular situations.

** Keywords. ** Fuzzy random variable, random set, variance, second order possibility measure

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** Authors addresses: **

Ines Couso

Department of Statistics and O.R.

E.U.I.T. Industrial de Gijon

Modulo 1-Planta 4

Campus Universitario de Viesques

33071 Gijon

Didier Dubois

Porte 308

IRIT, Université Paul Sabatier,

118 route de Narbonne,

31062, Toulouse, cedex 4, FRANCE.

Susana Montes

Department of Statistics and O.R.

E.U.I.T. Industrial de Gijon

Modulo 1-Planta 4

Campus Universitario de Viesques

33071 Gijon

Luciano Sanchez

Department of Computer Sciences

Edificio Departamental N1 - 1.1.28

Campus Universitario de Viesques

33071 Gijon

** E-mail addresses: **

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