On Distribution Characteristics of a Fuzzy Random Variable

Fuzzy Sets and Systems, 2006

This paper presents a backward analysis on the interpretation, modelling and impact of the concept of fuzzy random variable. After some preliminaries, the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained. We also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications.

Applied Mathematical Modelling, 2014

Characterizing the distribution of random elements is valuable for different purposes. Among them, inferential conclusions about the population distribution can be drawn on the basis of the sample one. When one deals with real-valued random variables this characterization is usually made through the distribution function or other ones, like the moment-generating or the characteristic functions. In case of dealing with random elements taking on fuzzy number values, the distribution function cannot be adequately defined in terms of a total ordering since there is no universally acceptable one for fuzzy numbers. This paper introduces a characterization of the distribution of these random elements by extending the moment-generating function. Properties of this extension are examined, and the notion is illustrated by means of some examples.

The notion of Fuzzy Random Variable has been introduced to model random mechanisms generating imprecisely-valued data which can be properly described by means of fuzzy sets. Probabilistic aspects of these random elements have been deeply discussed in the literature. However, statistical analysis of fuzzy random variables has not received so much attention, in spite that implications of this analysis range over many fields, including Medicine, Sociology, Economics, and so on. A summary of the fundamentals of fuzzy random variables is presented. Then, some related "parameters" associated with the distribution of these variables are defined. Inferential procedures concerning these "parameters" are described. Some recent results related to linear models for fuzzy data are finally reviewed.

International Journal of Theoretical Physics, 2002

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach to fuzzy probability theory (see [8, 10]), the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and present an example indicates that the converse is not true, we prove the strong law of large numbers according to this definition.

Kybernetika -Praha-

The concepts of cumulative distribution function and empirical distribution function are investigated for fuzzy random variables. Some limit theorems related to such functions are established. As an application of the obtained results, a method of handling fuzziness upon the usual method of Kolmogorov-Smirnov one-sample test is proposed. We transact the α-level set of imprecise observations in order to extend the usual method of Kolmogorov-Smirnov one-sample test. To do this, the concepts of fuzzy Kolmogorov-Smirnov one-sample test statistic and p-value are extended to the fuzzy Kolmogorov-Smirnov one-sample test statistic and fuzzy p-value, respectively. Finally, a preference degree between two fuzzy numbers is employed for comparing the observed fuzzy p-value and the given fuzzy significance level, in order to accept or reject the null hypothesis of interest. Some numerical examples are provided to clarify the discussions in this paper.

2008

Addition of two Fuzzy Bernoulli distribution and the sum of subsequent fuzzy binomial distributions have been discussed in this paper. Extensions of these ideas would be of use to study fuzzy randomness and the concept of measure.

Information Sciences, 2001

In this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable. This discussion is mainly focused on ®nding an adequate notion of measurability to be coherent with the notions on the space these random elements take values. Ó

Fuzzy Sets and Systems, 1998

The concepts of fuzzy random variable, and the associated fuzzy expected value, have been introduced by Purl and Ralescu as an extension of measurable set-valued functions (random sets), and of the Aumann integral of these functions, respectively. On the other hand, the 2-average function has been suggested by Campos and Gonzhlez as an appropriate function to rank fuzzy numbers. In this paper we are going to analyze some useful properties concerning the 2-average value of the expectation of a fuzzy random variable, and some practical implications of these properties are also commented on.

A parametric method has been developed to estimate the fuzzy mean and variance of fuzzy random variable using Gumbel distribution. It is an extreme value type-I distribution which is a limiting model for the maximum and minimum values of a data set. In order to obtain the fuzzy mean and variance, first, using the method of alpha cut, crisp intervals from each of the fuzzy observation for the same α level are generated. Thereafter, from these crisp intervals for the same α level, the probability distributions of the extreme values and its parameters have been estimated. The interval containing these two parameters as extreme points will represent the estimated crisp interval parameter of the crisp intervals for a certain α level. After estimating all the interval parameters for different levels of alpha cuts, the fuzzy mean and variance of a fuzzy random variable are estimated.

Journal of Mathematical Analysis and Applications, 1982

A general framework for a theory is presented that encompasses both statistical uncertainty. which falls within the province of probability theory, and nonstatistical uncertamty. which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. FUZZJ Sers I (1978). 3-281. The concept of a fuzzy integral ts used to define the expected value of a random vartable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence IS formulated, various convergence concepts are defined and their relationshtps are studied by using a Chebyshevlike inequality for the fuzzy Integral. The possibility of using this theory m Bayestan estimation with fuzzy prior mformation IS explored.