8
Fuzzy probability distributions
Standard probability densities are usually motivated by histograms. Based on fuzzy histograms from Chapter and by the fuzziness of a priori distributions in Bayesian inference a generalization of probability distributions is necessary. Moreover, a generalized law of large numbers for fuzzy valued sequences of random variables leads to so-called fuzzy probability densities.
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
Remark 8.1:
Fuzzy probability densities can be graphically displayed by drawing δ-level functions ...
