8.2 Probabilities based on fuzzy probability densities
Based on the calculation of probabilities of events for classical continuous probability distributions by integrals, i.e.
one could think to use the generalized integration from Definition 3.3 for fuzzy probability densities f*(·).
However, this is not suitable because
but as abstraction of the relative frequency hn*(M) for fuzzy data hn*(M) = 1 has to hold true.
Therefore another generalized integration is necessary. This is possible by introducing the so-called fuzzy probability integral
Let f*(·) be a fuzzy probability density defined on a measure space (M, , μ), and consider the set Sδ of all classical probability densities f (·) on (M, , μ) obeying for all x ∈ M.
Then the fuzzy probability P*(A) for A ∈ is the fuzzy ...