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Statistical Methods for Fuzzy Data by Reinhard Viertl

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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.

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one could think to use the generalized integration from Definition 3.3 for fuzzy probability densities f*(·).

However, this is not suitable because

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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

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Definition 8.2:

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 xM.

Then the fuzzy probability P*(A) for A is the fuzzy ...

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