ELSEVIER Statistics & Probability Letters 36 (1997) 135-143 STATISTICS& PROBABILITY LETTERS Constructive definitions of fuzzy random variables Miguel Lrpez-Diaz, Maria Angeles Gil* Departamento de Estadistica, Universidad de Oviedo, 33071 Oviedo, Spain Received 1 July 1996; received in revised form 1 February 1997 Abstract When we deal with a random experiment, we are often interested in functions of the experimental outcomes rather than the outcomes themselves. Fuzzy random variables formalize fuzzy-valued functions of the outcomes in a random experiment, that is, existing imprecise quantification processes. The concepts of fuzzy random variable and its fuzzy expected value, have been introduced by Puri and Ralescu by means of descriptive definitions. Nevertheless, constructive definitions of fuzzy random variables would play an essential role in the constructive definition of their integrals, which will be especially valuable to perform practical computations and to develop further results concerning the integration of these variables. In this paper we present constructive definitions of fuzzy random variables and integrably bounded fuzzy random variables based on the Hausdorff convergence. The use of the last definition to obtain a constructive definition of the fuzzy expected value of an integrably bounded fuzzy random variable is finally discussed. (~) 1997 Elsevier Science B.V. AMS classification: Primary 60D05, 03E72; secondary 28B20 Keywords: Random compact convex set; fuzzy random variable; integrably bounded fuzzy random variable; ~t-level func- tion; Hausdorff metric; Hausdorff convergence 1. Introduction Given a random experiment, and a probability space modeling this experiment, one traditionally assumes that the experimental performance is accomplished under randomness, and the quantification process associated with the experimental outcomes is real-valued. However, in practice fuzziness can arise in a quantification process, so that it assigns to each experimental outcome an imprecise value. To model such a process, Puri and Ralescu (1986), and Klement et al. (1986) (see also Ralescu, 1995), have introduced the concept of fuzzy random variable as an extension of both, random variables and random sets. The concept of fuzzy random variable has been introduced in a descriptive way, on the basis of the notion of measurability of certain set-valued functions (the level functions). This descriptive definition is suitable for purposes of analyzing and proving most of the properties of fuzzy random variable. Nevertheless, in the development of the integrals of these variables, as well as in the practical computation of these * Corresponding author. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S0167-71 52(97)00056-4