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Statistical Methods for Fuzzy Data by Reinhard Viertl

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25.2 Expectation and variance of fuzzy random variables

Let be a random quantity defined on (Ω, , P) with values in a metric space . Using the so-called Fréchet principle} the (not necessarily unique) expectation is defined to be the set of all AM, which minimize the squared distance , i.e.

(25.1) Numbered Display Equation

if an AM exists with (see Fréchet, 1948).

The infimum of the expected quadratic distance is called the Fréchet variance, i.e.

(25.2) Numbered Display Equation

Remark 25.3:

For classical random variables X we have

Unnumbered Display Equation

Therefore the Fréchet principle generalizes the classical variance.

The Fréchet principle for the expectation ...

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