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Statistical Methods for Fuzzy Data by Reinhard Viertl

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25.4 Further results

A fuzzy stochastic quantity x* in LR-form is defined by four real random variables m, s, l and r with and two fuzzy numbers

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In order to fulfill the random variables m, s, l and r have to be square-integrable. By the linearity of the expectation operator the expectation X* is a fuzzy number in LR-form obeying

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The variance of this fuzzy random quantity is given by

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Many results for real random variables can be generalized for fuzzy random quantities.

Proposition 25.2 (Chebysheff’s inequality):

For fuzzy stochastic quantities X* with < ∞ the following holds:

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Proof:

By the ...

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