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On the variance of fuzzy random variables. (English) Zbl 0936.60017

Frechét has defined an expectation of a random fuzzy variable (rfv) \(X\) with values in a metric space \((M,d)\) by that \(a^*\in M\) which minimizes \(Ed^2(X,a)\) and he has further defined \(\text{Var} X:=Ed^2(X,a^*)\). In the present paper it is shown that the Aumann-expectation of a rfv is Frechét w.r.t. the \(L^2\)-metric between support functions, associated with convex rfv’s. Then the Frechét-principle leads to an appropriate variance of rfv’s. The author discusses properties of that variance and presents special formulas for random \(LR\)-fuzzy-numbers. As application, best linear estimation and best linear prediction in linear regression models with fuzzy data and a strong law of large numbers w.r.t. the used \(L^2\)-metric is considered.

MSC:

60E99 Distribution theory
60A99 Foundations of probability theory
03E72 Theory of fuzzy sets, etc.
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