9
A law of large numbers
Random samples whose values are fuzzy exhibit also a kind of convergence in extension of the classical law of large numbers. A suitable mathematical model for random variables which assume fuzzy values are so-called fuzzy random variables. Based on this concept a generalized law of large numbers can be formulated.
9.1 Fuzzy random variables
Let (Ω, 
, P) be a classical probability space. The generalization of random variables X defined on (Ω, , P), i.e. X is a Borel-measurable function X: Ω → 
, to the situation of fuzzy valued quantities are fuzzy random variables, which are defined in the following way.
Definition 9.1:
A fuzzy random variable X* on a probability space (Ω, , P) is a function from Ω to the set of fuzzy numbers 
such that
for every Borel-set B and every δ ∈[0; 1], where Xδ(ω) denotes the δ-cut of X*(ω).
Remark 9.1:
If the values X*(ω) are all fuzzy ...
