A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing

A family of fuzzy representations of random variables is presented. Each representation transforms a real-valued random variable into a fuzzy-valued one. These representations can be chosen so that they lead to fuzzy random variables whose means capture different relevant information on the probability distribution of the original real-valued random variable. In this way, the means of the transformed fuzzy random variables can capture, for instance, immediate visual information about some key parameters, and even the whole information about the distribution of the original random variable. Representations capturing visual information on parameters of the original random variable may be considered for statistical descriptive/exploratory purposes. Representations for which the fuzzy mean characterizes the distribution of the original random variable will be mainly valuable to develop statistical inferences on this variable. Some interesting inferential applications for classical random variables based on the last fuzzy representations are commented, and an example illustrates one of them empirically and motivate future directions and discussions.