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 A ∈ M, which minimize the squared distance , i.e.
if an A ∈ M exists with (see Fréchet, 1948).
The infimum of the expected quadratic distance is called the Fréchet variance, i.e.
For classical random variables X we have
Therefore the Fréchet principle generalizes the classical variance.
The Fréchet principle for the expectation ...