On fuzzy random variables and statistical maps

An apparently natural concept of fuzzy random variables is introduced and discussed. Its relation to the standard random variables as well as to the affine mappings between the convex sets of all probability measures on the respective measurable spaces is investigated; in particular, it is shown that the standard random variables are in one-to-one correspondence to the extreme points of the convex set of all fuzzy random variables with the same outcome space. Moreover, some physical situations are considered in which the fuzzy random variables presented here provide the appropriate mathematical description.