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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 14

a. Fuzzy discrete probability distributions on finite sets {x1,…, xn} are given by their fuzzy point probabilities p*(xj) which are fuzzy intervals. For a subset {xj1,…,xjm} of {x1,…,xk} the probability P*({xj1,…,xjm}) is given via its δ-cuts Cδ(P*({xj1,…,xjm}) which are defined by all possible sums of numbers ylCδ(p*(xjl)) with

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By this construction condition (1) from Definition 8.3 is fulfilled. Condition (2) follows from the assumption that is a fuzzy interval with

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In order to prove condition (3) consider pair wise disjoint subsets A1,…,Am of {x1,…,xk}. Denoting the value is obtained as the maximum of all sums with ylCδ(p*(xjl) where = 1 . On the other hand δ(Ai) is ...

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