Abstract
Fuzzy random variable has been defined in several ways in literature. This paper presents a new definition of fuzzy random variable, and gives a novel definition of scalar expected value operator for fuzzy random variables. Some properties concerning the measurability of fuzzy random variable are also discussed. In addition, the concept of independent and identically distributed fuzzy random variables is introduced. Finally, a type of law of large numbers is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aumann, R. J. (1965). “Integrals of Set-Valued Functions,” J. Math. Anal. Appl. 12, 1–12.
Butnariu, D. (1989). “Measurability Concepts for Fuzzy Mappings,” Fuzzy Sets and Systems 31, 77–82.
Dubois, D. and H. Prade. (1988). Possibility Theory. New York: Plenum Press.
Himmelberg, C. J. (1975). “Measurable Relations,” Fund. Math. 87, 53–72.
Kaleva, O. (1987). “Fuzzy Differential Equations,” Fuzzy Sets and Systems 24, 301–317.
Kallenberg, O. (1997). Foundations of Modern Probability. Springer-Verlag.
Klement, E. P., M. L. Puri, and D. A. Ralescu. (1986). “Limit Theorems for Fuzzy Random Variables,” Proc. R. Soc. Lond. A 407, 171–182.
Klir, G. J. (1999). “On Fuzzy-Set Interpretation of Possibility Theory,” Fuzzy Sets and Systems 108, 263–273.
Kruse, R. (1982). “The Strong Law of Large Numbers for Fuzzy Random Variables,” Information Sciences 28, 233–241.
Kruse, R. and K. D. Meyer. (1987). Statistics with Vague Data. Dordrecht: D. Reidel Publishing Company.
Kwarkernaak, H. (1978). “Fuzzy Random Variables—I,” Information Sciences 15, 1–29.
Kwarkernaak, H. (1979). “Fuzzy Random Variables—II,” Information Sciences 17, 153–178.
Liu, B. (1999). Uncertain Programming. New York: John Wiley & Sons.
Liu, B. (2001a). “Fuzzy Random Chance-Constrained Programming,” IEEE Transactions on Fuzzy Systems 9, 713–720.
Liu, B. (2001b). “Fuzzy Random Dependent-Chance Programming,” IEEE Transactions on Fuzzy Systems 9, 721–726.
Liu, B. and Y.-K. Liu. (2002). “Expected Value of Fuzzy Variable and Fuzzy Expected Value Models,” IEEE Transactions on Fuzzy Systems 10, 445–450.
Liu, Y.-K. (2000). “On the Convergence of Measurable Set–Valued Function Sequence on Fuzzy Measure Space,” Fuzzy Sets and Systems 112, 241–249.
Luhandjula, M. K. (1996). “Fuzziness and Randomness in an Optimization Framework,” Fuzzy Sets and Systems 77, 291–297.
Luhandjula, M. K. and M. M. Gupta. (1996). “On Fuzzy Stochastic Optimization,” Fuzzy Sets and Systems 81, 47–55.
Negoita, C. V. and D. Ralescu. (1987). Simulation, Knowledge-Based Computing, and Fuzzy Statistics. New York: Van Nostrand Reinhold Company.
Puri, M. L. and D. A. Ralescu. (1985). “The Concept of Normality for Fuzzy Random Variables,” Annals of probability 13, 1371–1379.
Puri, M. L. and D. A. Ralescu. (1986). “Fuzzy Random Variables,” J. Math. Anal. Appl. 114, 409–422.
Yazenin, A. V. (1987). “Fuzzy and Stochastic Programming,” Fuzzy Sets and Systems 22, 171–180.
Zadeh, L. A. (1978). “Fuzzy Sets As a Basis for a Theory of Possibility,” Fuzzy Sets and Systems 1, 3–28.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, YK., Liu, B. Fuzzy Random Variables: A Scalar Expected Value Operator. Fuzzy Optimization and Decision Making 2, 143–160 (2003). https://doi.org/10.1023/A:1023447217758
Issue Date:
DOI: https://doi.org/10.1023/A:1023447217758