Descriptive statistics with vague data

Abstract

The study of probability distributions of a random variable is essentially the study of some numerical characteristics associated with them. These so-called parameters of the distribution play a key role in mathematical statistics. We generalize these methods and we define the notions of moments of fuzzy random variables. The reason for considering moments is that one is often interested in a short description of a population. This can be achieved with statistical characteristics. There are a lot of useful parameters defined in the literature on statistics. To answer the question ”What is the ’centre’ of the observed values x 1 ,...,x n ? ”, we can use location parameters, such as arithmetic mean, geometric mean, harmonic mean, modal value, median, etc. It depends on the situation which parameter is appropriate. Since questions of this kind have been answered for those cases in which only randomness is involved, we can restrict ourselves to consider the further problem that arises if randomness and vagueness appear simultaneously. It follows then that we are able to extend the known results of random variables to random sets and to fuzzy random variables. We will concentrate on moments and their properties.