Upper and lower probabilities induced by a fuzzy random variable

We review two existing interpretations of fuzzy random variables. In the first one, the fuzzy random variable is viewed as a linguistic random variable. In the second case, it represents some incomplete knowledge about an otherwise standard random variable. For each interpretation, the information provided by the frv is described by a specific model, namely a standard probability model and a second-order imprecise model, respectively. In this paper, we deal with an alternative interpretation. Guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations. Then we will demonstrate that the new interpretation leads, in a natural way, to the pair of order ∞ capacities. Furthermore, we show that they are formally related to the former models. These results can be applied in future works to make inferences from fuzzy sample data. The use of upper–lower models instead of second-order models will enable us in the future to reach crisp decisions in some specific statistical problems, without adding any arbitrary information, and taking into account the imprecision in data.