## 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:

 [1] Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301 [2] 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. [3] 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 [4] 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. [5] Cox, E., The fuzzy systems handbook, (1994), Academic Press Cambridge [6] 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 [7] Körner, R., On the variance of fuzzy random variables, Fuzzy sets and systems, 92, 83-93, (1997) · Zbl 0936.60017 [8] 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 [9] 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 [10] 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. [11] Näther, W., Linear statistical inference for random fuzzy data, Statistics, 29, 221-240, (1997) · Zbl 1030.62530 [12] Negoita, C.V.; Ralescu, D.A., Simulation, knowledge-based computing, and fuzzy statistics, (1987), Van Nostrand Reinhold New York · Zbl 0683.68097 [13] Puri, .L.; Ralescu, D.A., Différentielle d’une fonction floue, C.R. acad. sci. Paris Sér. A, 293, 237-239, (1981) · Zbl 0489.46038 [14] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.