Fuzzy Random Variables: Modeling Linguistic Statistical Data

  • María Angeles Gil
  • Pedro A. Gil
  • Dan A. Ralescu
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 34)


This paper deals with a concept which connects Probability and Fuzzy Set Theories: the fuzzy random variable. Fuzzy random variables represent an operational and rigorous model to formalize linguistic variables associated with numerical quantification processes (like measurements or counting). This paper offers a review of Puri and Ralescu’s definition of fuzzy random variables and gathers most of the probabilistic and statistical results and methods based on it.


Fuzzy Number Probability Space Linguistic Variable Fuzzy Subset Finite Population 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artstein, Z. and Vitale, R.A. (1975). A strong law of large numbers for random compact sets. Ann. Probab. 3 879–882.MathSciNetMATHCrossRefGoogle Scholar
  2. Aumann, R.J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 121–12.MathSciNetMATHCrossRefGoogle Scholar
  3. Bertoluzza, C., Corral, N. and Salas, A. (1995). On a new class of distances between fuzzy numbers. Mathware & Soft Computing 2 71–84.MathSciNetMATHGoogle Scholar
  4. Bochner, S. (1933). Integration von funktionen, deren werte die elemente eines vektorraumes sind. Fundamenta Math. 20 262–176.Google Scholar
  5. Byrne, C. (1978). Remarks on the set-valued integrals of Debreu and Aumann. J. Math. Anal. Appl. 78 243–246.MathSciNetCrossRefGoogle Scholar
  6. Colubi, A., López-Diaz, M., Domínguez-Menchero, J.S. and Gil, M.A. (1997). A generalized Strong Law of Large Numbers. Thch. Rep. University of Oviedo, June 1997.Google Scholar
  7. Debreu, G. (1966). Integration of correspondences. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 351–372. Univ of California Press, Berkeley.Google Scholar
  8. Gil, M.A. and López-Diaz, M. (1996). Fundamentals and Bayesian Analyses of decision problems with fuzzy-valued utilities. Int. J. Approx. Reason. 15 203–224.MATHCrossRefGoogle Scholar
  9. Gil, M.A., López-Diaz, M. and López-Garcia, H. (1997a). The fuzzy hyperbolic inequality index associated with fuzzy random variables. European J Oper. Res. (in press).Google Scholar
  10. Gil, M.A., López-Díaz, M. and Rodriguez-Muíïiz, L.J. (1997b). An improvement of a comparison of experiments in statistical decision problems with fuzzy utities. IEEE Trans. Syst. Man, Cyb. (in press).Google Scholar
  11. Klement, E.P., Puri M.L. and Ralescu, D.A. (1986) Limit theorems for fuzzy random variables. Proc. R. Soc, Lond. A 407, 171–182.Google Scholar
  12. López-Díaz, M. and Gil, M.A. (1997a). Constructive definitions of fuzzy random variables. Stat. Probab. Lett. (in press).Google Scholar
  13. López-Díaz, M. and Gil, M.A. (1997b). Approximating integrably bounded fuzzy random variables in terms of the “generalized” Hausdorff metric. Inform. Sci. (in press).Google Scholar
  14. López-Díaz, M. and Gil, M.A. (1997c). Reversing the order of integration in iterated expectations of fuzzy random variables, and some statistical applications. Thch. Rep. University of Oviedo, January 1997.Google Scholar
  15. López-García, H. (1997). Cuantificación de la desigualdad asociada a conjuntos aleatorios y variables aleatorias difusas. PhD Thesis, University of Oviedo.Google Scholar
  16. López-García, H., Gil, M.A., Corral, N. and López, M.T. (1997). Estimating the fuzzy inequality associated with a fuzzy random variable in random samplings from finite populations. Kybernetika (in press).Google Scholar
  17. Lubiano, A., Gil, M.A., López-Díaz, M. and López, M.T. (1997). The real-valued mean squared dispersion associated with a fuzzy random variable. Téch. Rep. University of Oviedo, April 1997.Google Scholar
  18. Lubiano, A. and Gil, M.A. (1997). Estimating the expected value of fuzzy random variables in random samplings from finite populations. Tech. Rep. University of Oviedo, July 1997.Google Scholar
  19. Puri, M.L. and Ralescu, D. (1981). Différentielle d’une fonction floue. C.R. Acad. Sci. Paris, Sér. I 293 237–239.MathSciNetMATHGoogle Scholar
  20. Puri, M.L. and Ralescu, D. (1985). The concept of normality for fuzzy random variables. Ann. Probab. 13 1373–1379.MathSciNetMATHCrossRefGoogle Scholar
  21. Puri, M.L. and Ralescu, D. (1986). Fuzzy random variables. J. Math. Anal. Appl. 114 409–422.MathSciNetMATHCrossRefGoogle Scholar
  22. Ralescu, A. and Ralescu, D.A. (1984). Probability and fuzziness. Information Sciences 17 85–92.MathSciNetCrossRefGoogle Scholar
  23. Ralescu, A. and Ralescu, D.A. (1986). Fuzzy sets in statistical inference. The Mathematics of Fuzzy Systems (A. Di Nola and A.G.S. Ventre, Eds.), Verlag TVV Rheinland, Köln, pp. 273–283.Google Scholar
  24. Ralescu, D.A. (1982). Fuzzy logic and statistical estimation. Proc. 2nd World Confer- ence on Mathematics at the Service of Man 605–606, Las Palmas-Canarias.Google Scholar
  25. Ralescu, D.A. (1995a). Fuzzy random variables revisited. Proc. IFES’95 and Fuzzy IEEE Joint Conference, Vol. 2 993–1000, Yokohama.Google Scholar
  26. Ralescu, D.A. (1995b). Inequalities for fuzzy random variables. Proc. 26th Iranian Mathematical Conference 333–335, Kerman.Google Scholar
  27. Ralescu, D.A. (1995c). Fuzzy probabilities and their applications to statistical inference. Advances in Intelligent Computing — IPMU’94, Lecture Notes in Computer Science 945 217–222.CrossRefGoogle Scholar
  28. Ralescu, D.A. (1996). Statistical Decision-Making without numbers. Proc. 27th Iranian Mathematical Conference 403–417, Shiraz.Google Scholar
  29. Zadeh, L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning. Parts 1,2, and 3. Inf. Sci. 8 199–249; 8 301–357; 9 43–80.Google Scholar
  30. Zadeh, L.A. (1976). A fuzzy-algorithmic approach to the definition of complex or imprecise concepts. Int. Jour. Man-Machine Studies, 8 249–291.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • María Angeles Gil
    • 1
  • Pedro A. Gil
    • 1
  • Dan A. Ralescu
    • 2
  1. 1.Departamento de Estadística, I.O. y D.M.Universidad de OviedoOviedoSpain
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations