## 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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### References:

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