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Statistical Methods for Fuzzy Data by Reinhard Viertl

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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

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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 intervals, then the set Xδ(ω) is a compact interval for all δ ∈ [0; 1], i.e.

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Xδ is then an interval-valued function.

Lemma 9.1:

Let (Ω, , P) ...

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