## The $$\vec \lambda$$-mean squared dispersion associated with fuzzy random variable.(English)Zbl 0973.60005

The framework of the paper is based on the notions of fuzzy random variable and fuzzy expected value of a fuzzy random variable, introduced by M. I. Puri and D. A. Ralescu [J. Math. Anal. Appl. 114, 409-422 (1986; Zbl 0592.60004)]. This paper defines a parameterized real-valued measure of the mean dispersion of a fuzzy random variable with respect to an arbitrary fuzzy number. The proposed measure extends the second moment of a classical random variable, and it is based on a parameterized distance between fuzzy numbers. Properties of the measure presented are examined, including the extension of the variance of a classical random variable, and its particular case as a mean squared dispersion. Some examples illustrating the proposed measure within the computation and use of the fuzzy random variables are enclosed. Other examples of its application are just mentioned: (a) the evaluation of the precision of the sample expected value of a fuzzy random variable in estimating the population value, when a random sampling from a finite population is considered, and (b) estimation of fuzzy parameters for fuzzy random variables in a Bayesian context.

### MSC:

 60A99 Foundations of probability theory 28E10 Fuzzy measure theory

Zbl 0592.60004
Full Text:

### References:

  Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301  C. Bertoluzza, N. Corral, A. Salas, Fuzzy Linear Regression: existence of solution for a generalized least squares method, Proc. 4th IFSA Conf., CMS, Brussels, 1991, pp. 233-235.  Bertoluzza, C.; Corral, N.; Salas, A., On a new class of distances between fuzzy numbers, Mathware soft comput., 2, 71-84, (1995) · Zbl 0887.04003  C. Bertoluzza, N. Corral, A. Salas, Polynomial regression in a fuzzy context. The least squares method, Proc. 6th IFSA Conf., Sao Paolo, 1995, pp. 431-434.  Cox, E., The fuzzy systems handbook, (1994), Academic Press Cambridge  Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, Proc. roy. soc. London ser. A, 19, 171-182, (1986) · Zbl 0605.60038  Körner, R., On the variance of fuzzy random variables, Fuzzy sets and systems, 92, 83-93, (1997) · Zbl 0936.60017  López-Dı́ az, M.; Gil, M.A., The $$λ$$-average value of the expected value of a fuzzy random variable, Fuzzy sets and systems, 99, 347-352, (1998) · Zbl 0941.60030  A. Lubiano, M.A. Gil, Estimating the expected value of fuzzy random variables in random samplings from finite populations, Tech. Rep., University of Oviedo, July 1997. · Zbl 0942.62010  M. Montenegro, M.R. Casals, M.A. Gil, Estimating fuzzy parameters of fuzzy random variables in a Bayesian context, Proc. IPMU’98 Conf., 1998, in press.  Näther, W., Linear statistical inference for random fuzzy data, Statistics, 29, 221-240, (1997) · Zbl 1030.62530  Negoita, C.V.; Ralescu, D.A., Simulation, knowledge-based computing, and fuzzy statistics, (1987), Van Nostrand Reinhold New York · Zbl 0683.68097  Puri, .L.; Ralescu, D.A., Différentielle d’une fonction floue, C.R. acad. sci. Paris Sér. A, 293, 237-239, (1981) · Zbl 0489.46038  Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
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