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Fuzzy random variables—I. definitions and theorems

Huibert Kwakernaak
- 01 Jul 1978 - 
- Vol. 15, Iss: 1, pp 1-29
TLDR
Fuzziness is discussed in the context of multivalued logic, and a corresponding view of fuzzy sets is given.
About: 
This article is published in Information Sciences.The article was published on 1978-07-01 and is currently open access. It has received 1161 citations till now. The article focuses on the topics: Fuzzy number & Fuzzy set operations.

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Citations
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Journal ArticleDOI

Fuzzy Random Variables

Madan L. Puri, +1 more
- 01 Mar 1986 - 
TL;DR: The new definition of expectation generalizes the integral of a set-valued function and derives the Lebesgue-dominated convergence type theorem by considering a suitable generalization of the Hausdorff metric.
Book

Theory and practice of uncertain programming

Baoding Liu
TL;DR: This book provides a self-contained, comprehensive and up-to-date presentation of uncertain programming theory, including numerous modeling ideas, hybrid intelligent algorithms, and applications in system reliability design, project scheduling problem, vehicle routing problem, facility location problem, and machine scheduling problem.
Journal ArticleDOI

The mean value of a fuzzy number

Didier Dubois, +1 more
- 01 Dec 1987 - 
TL;DR: Some light is shed on the debate between probability and possibility theory, and the problem of how to carry measure-theoretic notions into the field of possibility theory.

Fuzzy Process, Hybrid Process and Uncertain Process

Baoding Liu
TL;DR: In order to construct fuzzy counterparts of Brownian motion and stochastic calculus, some basic concepts of fuzzy process are proposed, including fuzzy calculus and fuzzy difierential equation, which are extended to hybrid process and uncertain process.
Book

Ignorance and Uncertainty: Emerging Paradigms

Michael Smithson
TL;DR: In this article, a variety of approaches to the problem of indeterminacies in human thought and behavior are discussed, including cognitive psychology, social psychology, organizational studies, sociology, and social anthroplogy.
References
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Journal ArticleDOI

The concept of a linguistic variable and its application to approximate reasoning—II☆

Lotfi A. Zadeh
- 01 Jan 1975 - 
TL;DR: Much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.
Book

Stochastic processes

J. L. Doob, +1 more
Book

Probability theory

Michel Loève
TL;DR: These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of large numbers.
Book

Probability Theory I

Michel Loève
Journal ArticleDOI

Probability measures of Fuzzy events

Lotfi A. Zadeh
- 01 Aug 1968 - 
TL;DR: In probability theory, an event, A, is a member of a a-field, CY, of subsets of a sample space ~2, where CY is any collection of disjoint events.
Frequently Asked Questions (6)
Q1. What are the contributions in this paper?

In this paper, the notion of random variables is extended to fuzzy random variables, allowing impreciseness in the values that are assumed by the random variable. 

Because of tire central role played by the fuzzy set X=(%,X), the authors adopt in the following the convention of calling X a fuzzy random variable, as an alternative to calling the map 6 a fuzzy random variable. 

Denoting the membership function of the fuzzy number EX as (EX) (a notation that will frequently be used), the authors may explicitly write EX= (R, (EX)), where(EX)(x) = sup inf X, ( 0 (a, W’)), XfR. fi&:Eil~X $gj<(54A fuzzy random variable X is called z&n&l if for each w E 51, the membership function X, is unimodal. 

The condition that for each ~E(O, I] both VJ and U;* are finite random variables constitutes a restriction that from a practical point of view is not very serious. 

Properties of fuzzy random variables such as its expectation and probabilities in connection with it will be defined as images of this fuzzy set under certain mappings. 

Property (b) expresses that t(aAb) and t(ai,/b) do not become less true if a is changed so that its truth value increases, and that moreover the dependence is continuous.