A Note on Fuzzy Probability of a Fuzzy Event

This article modifies evaluating the probability of a fuzzy event based on a classical probability space introduced by Yager (Inform. Sci. 1979;18:113–122). The presented theoretical results will be illustrated with some propositions and remarks. It is shown that a modification in the definition of fuzzy probability of fuzzy events into a concept of such probabilities has some desirable properties from mathematical points of view. Then, the proposed fuzzy probability will be compared to that of the similar proposed methods. The proposed fuzzy probability will be illustrated with some numerical examples.

[1]  Mila Stojakovic.  Imprecise set and fuzzy valued probability , 2011, J. Comput. Appl. Math..

[2]  Zhong You-ming,et al.  Random variable with fuzzy probability , 2003 .

[3]  Stanisław Heilpern.  Fuzzy subsets of the space of probability measures and expected value of fuzzy variable , 1993 .

[4]  Jonathan Lawry,et al.  A mass assignment theory of the probability of fuzzy events , 1996, Fuzzy Sets Syst..

[5]  R. Yager.  A representation of the probability of a fuzzy subset , 1984 .

[6]  Fuzzy probability space and extension theorem , 1986 .

[7]  Herbert Toth,et al.  Probabilities and fuzzy events: an operational approach , 1992 .

[8]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[9]  Lin Liu,et al.  Fuzzy probability measures (FPM) based non-symmetric membership function: Engineering examples of ground subsidence due to underground mining , 2010, Eng. Appl. Artif. Intell..

[10]  Yian-Kui Liu,et al.  Expected Value Operator of Random Fuzzy Variable, Random Fuzzy Expected Value Models , 2003, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[11]  Thomas Augustin,et al.  Nonparametric predictive inference and interval probability , 2004 .

[12]  Michael Beer,et al.  A Summary on Fuzzy Probability Theory , 2010, 2010 IEEE International Conference on Granular Computing.

[13]  William E. Stein.  Fuzzy probability vectors , 1985 .

[14]  Matthias C. M. Troffaes.  Decision making under uncertainty using imprecise probabilities , 2007, Int. J. Approx. Reason..

[15]  Edward K. Wong,et al.  Fuzzy Independence and Extended Conditional Probability , 1996, Inf. Sci..

[16]  Erich-Peter Klement,et al.  Some remarks on a paper by R. R. Yager , 1982, Inf. Sci..

[17]  P. Smets.  Probability of a fuzzy event: An axiomatic approach , 1982 .

[18]  M. Puri,et al.  Convergence theorem for fuzzy martingales , 1991 .

[19]  Jing-Shing Yao,et al.  Fuzzy probability over fuzzy σ-field with fuzzy topological spaces , 2000, Fuzzy Sets Syst..

[20]  Giulianella Coletti,et al.  Conditional probability and fuzzy information , 2006, Comput. Stat. Data Anal..

[21]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[22]  Ljiljana Gajic,et al.  Fuzzy valued probability , 2015, Inf. Sci..

[23]  Beloslav Riecan,et al.  On a problem of Radko Mesiar: General form of IF-probabilities , 2006, Fuzzy Sets Syst..

[24]  Kwang Hyung Lee,et al.  First Course on Fuzzy Theory and Applications , 2005, Advances in Soft Computing.

[25]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[26]  Hsien-Chung Wu,et al.  Statistical hypotheses testing for fuzzy data , 2005, Inf. Sci..

[27]  Ronald R. Yager,et al.  A note on probabilities of fuzzy events , 1979, Inf. Sci..

[28]  C. Manski.  Partial Identification of Probability Distributions , 2003 .

[29]  Zhongyong Xia,et al.  Fuzzy probability system: fuzzy-probability space (1) , 2001, Fuzzy Sets Syst..

[30]  Z. Zenn Bien,et al.  INTEGRATED ADAPTIVE FUZZY CLUSTERING (IAFC) NEURAL NETWORKS USING FUZZY LEARNING RULES , 2005 .

[31]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[32]  Kevin D. Reilly,et al.  Simulating continuous fuzzy systems , 2005, Inf. Sci..

[33]  N. Singpurwalla,et al.  Membership Functions and Probability Measures of Fuzzy Sets , 2004 .

[34]  Jana Talašová,et al.  Fuzzy Probability Spaces and Their Applications in Decision Making , 2016 .

[35]  Krzysztof Piasecki.  Probability of fuzzy events defined as denumerable additivity measure , 1985 .

[36]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[37]  Charles F. Manski,et al.  Partial Identification of Probability Distributions (Springer Series in Statistics) , 2011 .