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Statistical Methods for Fuzzy Data by Reinhard Viertl

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8.1 Fuzzy probability densities

The theoretical counterparts of fuzzy histograms are fuzzy valued functions defined on observation spaces M, which are obeying the rules for fuzzy histograms (cf. Sections 3.6 and 5.2). These functions are called fuzzy probability densities (defined below).

Definition 8.1:

A fuzzy valued function defined on a measure space (M, , μ) is called a fuzzy probability density if it fulfills the following conditions:

1. The integral defined in Section 3.6 exists.

2. 1 ∈ C1(y*) and supp(y*) ⊆ (0, ∞).

3. There exists a classical probability density f : M → [0; ∞) with

Unnumbered Display Equation

Remark 8.1:

Fuzzy probability densities can be graphically displayed by drawing δ-level functions and of f*(·). An example of a one-dimensional fuzzy probability density, defined on the measure space (, , λ), with λ denoting the Lebesgue measure, is depicted in Figure 8.1.

Figure 8.1 One-dimensional ...

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