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Fuzzified probability : from Kolmogorov to Zadeh and beyond

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We discuss the fuzzification of classical probability theory. In particular, we point out similarities and differences between the so-called fuzzy probability theory and the so-called operational probability theory.
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autor
  • Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovak Republic
  • Catholic University in Ružomberok, Hrabovská cesta 1, 034 01 Ružomberok, Slovak Republic
autor
  • Catholic University in Ružomberok, Hrabovská cesta 1, 034 01 Ružomberok, Slovak Republic
  • Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovak Republic
Bibliografia
  • [1] M. Beer, Fuzzy Probability Theory, In: Encyclopedia of Complexity and Systems Science, volume 6 (Editor-in-chief: R. A. Meyers), Springer, New York, 2009, 4047-4059.
  • [2] S. Bugajski, Statistical maps I. Basic properties, Math. Slovaca 51 (2001), 321-342.
  • [3] S. Bugajski, Statistical maps II. Operational random variables, Math. Slovaca 51 (2001), 343-361.
  • [4] A. Colubi, J. S. Dominguez-Menchero, M. Lopez-Diaz, D. A. Ralescu, On the formalization of fuzzy random variables, Inform. Sci. 133 (2001), 3-6.
  • [5] R. Frič, Remarks on statistical maps and fuzzy (operational) random variables, Tatra Mountains Mathematical Publ. 30 (2005), 21-34.
  • [6] R. Frič, Extension of domains of states, Soft Comput. 13 (2009), 63-70.
  • [7] R. Frič, M. Papčo, A categorical approach to probability theory, Studia Logica 94 (2010), 215-230.
  • [8] R. Frič, M. Papčo, On probability domains II, Internat. J. Theoret. Phys. 50 (2011), 3778-3786.
  • [9] R. Frič and M. Papčo, Fuzzification of crisp domains, Kybernetika 46 (2010), 1009-1024.
  • [10] R. Frič, M. Papčo, Statistical maps and generalized random walks, Math. Slovaca 62 (2012), 1079-1090.
  • [11] S. Gudder, Fuzzy probability theory, Demonstratio Math. 31 (1998), 235-254.
  • [12] A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933. (An English translation by N. Morrison appeared under the title Foundations of the Theory of Probability (Chelsea, New York) in 1950.)
  • [13] H. Kwakernaak, Fuzzy random variables I. definitions and theorems. Inf. Sci. 15 (1978), 1-19.
  • [14] R. Mesiar, Fuzzy sets and probability theory, Tatra Mountains Mathematical Publ. 1 (1992), 105-123.
  • [15] M. Papčo, On measurable spaces and measurable maps, Tatra Mountains Mathematical Publ. 28 (2004), 125-140.
  • [16] M. Papčo, On fuzzy random variables: examples and generalizations, Tatra Mountains Mathematical Publ. 30 (2005), 175-185.
  • [17] M. Papčo, On effect algebras, Soft Comput. 12 (2007), 26-35.
  • [18] M. R. Puri, D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (2007), 409-422.
  • [19] L. A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968), 421-427.
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