Extremal properties of means of fuzzy random variables

In this paper, we examine extremal properties of fuzzy expectations and expectations of fuzzy random variables. We introduce a new mean characteristic—a scalar random variable that characterizes a given fuzzy random variable—and prove its extremal properties. Also, we study linear regressions of fuzzy random variables, obtain a formula for the optimal linear fuzzy regression, and prove that its correlation with the predicted value is maximal.