Qualitative Axioms Within Probability Calculus and Their Role in Comparing Previsions of Random Quantities

In this paper, the notion of probability is not undefined, so we extend to multilinear indices qualitative axioms which are not in conflict with those characterizing the development of modern probability theory: this is the main objective achieved by the current paper. The logical foundation of probability calculus is here extended by studying the mathematical expectation of random variables having two or more marginal variables as their components. Random variables, studied along with their probability distributions of a nonparametric nature, are geometric entities of which fundamental invariance properties are made explicit. The research gap addressed by this paper is the following: since the Cartesian product of two or more sets, where each of them is the image of a marginal random variable, is not commutative, noncommutative geometric objects coinciding with tensors come into play to make previsions of entities treating high-dimensional data. Empirical data given by time series of a finite length are handled. Time series of a finite length are formally seen as frequency distributions. They are also random variables. Hence, frequency distributions and random variables are shown to be the two sides of the same coin.