On various definitions of the variance of a fuzzy random variable

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 take some further decisions. In this work, we introduce a new 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 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.