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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 22

a. The fuzzy valued function

has δ-level functions

Unnumbered Display Equation

By assumption the integral exists and is finite for all δ ∈ (0; 1].

Therefore also

Unnumbered Display Equation

Now the fuzzy marginal density f*(·) is defined by its δ-level functions δ(·) and δ(·) respectively. Furthermore by Therefore f*(·) is a fuzzy density.

b. Let f(·) be a classical probability density with existing expectation, i.e. The δ-level functions of this density are all equal to f(·). Using the definition of Section 22.1 for the expectation we obtain

The characterizing function ψ (·) of μ* is given by the family of single point sets

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