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Constructive definitions of fuzzy random variables

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  • López-Diaz, Miguel
  • Gil, Maria Angeles

Abstract

When we deal with a random experiment, we are often interested in functions of the experimental outcomes rather than the outcomes themselves. Fuzzy random variables formalize fuzzy-valued functions of the outcomes in a random experiment, that is, existing imprecise quantification processes. The concepts of fuzzy random variable and its fuzzy expected value, have been introduced by Puri and Ralescu by means of descriptive definitions. Nevertheless, constructive definitions of fuzzy random variables would play an essential role in the constructive definition of their integrals, which will be especially valuable to perform practical computations and to develop further results concerning the integration of these variables. In this paper we present constructive definitions of fuzzy random variables and integrably bounded fuzzy random variables based on the Hausdorff convergence. The use of the last definition to obtain a constructive definition of the fuzzy expected value of an integrably bounded fuzzy random variable is finally discussed.

Suggested Citation

  • López-Diaz, Miguel & Gil, Maria Angeles, 1997. "Constructive definitions of fuzzy random variables," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 135-143, December.
    • Handle: RePEc:eee:stapro:v:36:y:1997:i:2:p:135-143
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    References listed on IDEAS

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    1. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
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    Cited by:

    1. Coppi, Renato & D'Urso, Pierpaolo & Giordani, Paolo & Santoro, Adriana, 2006. "Least squares estimation of a linear regression model with LR fuzzy response," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 267-286, November.
    2. Rodriguez-Muniz, Luis J. & Lopez-Diaz, Miguel & Gil, Maria Angeles, 2005. "Solving influence diagrams with fuzzy chance and value nodes," European Journal of Operational Research, Elsevier, vol. 167(2), pages 444-460, December.
    3. Gil, Maria Angeles & Lopez-Diaz, Miguel & Lopez-Garcia, Hortensia, 1998. "The fuzzy hyperbolic inequality index associated with fuzzy random variables," European Journal of Operational Research, Elsevier, vol. 110(2), pages 377-391, October.
    4. M. Gil & M. López-García & M. Lubiano & Manuel Montenegro, 2001. "Regression and correlation analyses of a linear relation between random intervals," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(1), pages 183-201, June.
    5. Volker Krätschmer, 2006. "Integrals of random fuzzy sets," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 433-469, September.
    6. Katagiri, Hideki & Sakawa, Masatoshi & Kato, Kosuke & Nishizaki, Ichiro, 2008. "Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability," European Journal of Operational Research, Elsevier, vol. 188(2), pages 530-539, July.
    7. Terán, Pedro, 2003. "A strong law of large numbers for random upper semicontinuous functions under exchangeability conditions," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 251-258, November.

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