By Irwin R. Goodman, Hung T. Nguyen (auth.), Professor Carlo Bertoluzza, Professor María-Ángeles Gil, Professor Dan A. Ralescu (eds.)

The contributions during this publication attach chance Theory/Statistics and Fuzzy Set conception in several methods. a few of these connections are both philosophical or theoretical in nature, yet so much of them country versions and techniques to paintings with fuzzy information (or fuzzy belief) whilst facing random experiments. during this manner, a number of probabilistic reviews are built, in addition to innovations and standards to get descriptive and inferential statistical conclusions from fuzzy information. nevertheless, a few stories were dedicated to fuzzy measures and their dating with measures in likelihood Theory.

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For example, (Fe (Hl,d) , d l ) appears as a complete and separable metric space, (Fe (Hl,d) , doo), however, is a complete but non-separable metric space (see Puri and Ralescu, 1986). Another type of distances can be defined via so called support functions. , . > is the scalar product in Hl,d and Sd-I the (d - I)-dimensional unit sphere in Hl,d. Note that for convex and compact A C Hl,d the support function SA is uniquely determined. A fuzzy set A E Fe(Hl,d) can be characterized a-cut-wise by its support function: a E (0,1]' U E Sd-I .

They defined a distance between two normal convex fuzzy sets A and B of the real line lR 1 by D(A, B)2 = 11 11 [t(inf Aex-inf Bex)+(l-t)(sup Aex-sup Bex)]2dg(t) d

Is the usual expectation of the real-valued variable d(Y,a? The variance of Z, denoted by Var(d)Z, is then defined by Var(d) Z = lEd(Z, lE(d) Z? (3) This is a generalization of the known fact that for a real valued random variable X the expectation lEX minimizes lElX _a1 2and Var X equals lElX - lEX12. t. d. For rfv Y, the Frechet approach opens the way for defining several expectations and their (via (3)) associated variances, each induced by a given metric between fuzzy sets. Therefore, first of all, we have to discuss on suitable distances between fuzzy sets.

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