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On fuzzy random variables and statistical maps. (English) Zbl 1026.60501
Summary: 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.

MSC:
60E99 Distribution theory
46N30 Applications of functional analysis in probability theory and statistics
46N50 Applications of functional analysis in quantum physics
81S25 Quantum stochastic calculus
28E10 Fuzzy measure theory
82B03 Foundations of equilibrium statistical mechanics
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