Overview on the Development of Fuzzy Random Variables 1 Mar´ ıa ´ Angeles Gil* a ,MiguelL´opez-D´ ıaz a and Dan A. Ralescu b a Departamento de Estad´ ıstica, I.O. y D.M., Universidad de Oviedo, Facultad de Ciencias, E-33071 Oviedo, Spain b Department of Mathematical Sciences University of Cincinnati, Cincinnati, OH 45221-0025, USA Abstract 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. Key words: d ∞ -metric, fuzzy arithmetic, fuzzy values, fuzzy random variables, Hausdorff metric, random sets, (W, ϕ)-metric. 1 Introduction Fuzzy random variables were introduced to model and analyze ‘imprecisely- valued’ measurable functions associated with the sample space of a random experiment, when the imprecision in values of these functions is formalized in terms of fuzzy sets. Different approaches to this concept have been developed in the literature, the most widely considered being that introduced by Kwakernaak (1978, 1979) and Kruse and Meyer (1987), and the one by Puri and Ralescu (1986) (and Klement, Puri and Ralescu, 1986). 1 The research in this paper has been partially supported by the Spanish Ministry of Education and Science Grants BFM2002-01057 and BFM2002-03263. This financial support is gratefully acknowledged. ∗ corresponding authors: magil@uniovi.es, Fax: (+34) 98 510 3356. Preprint submitted to Elsevier Science 8 July 2005