Abstract
There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributions.
Highlights
We propose a mathematical model where the probability of an event has a concrete image [1].On the other hand, the difference between two opposite points of view is well known: some scholars interpret probability as a subjective measure of the degree of belief, whereas others consider it as an objective measure connected with measurable sets [2]
It is analytically possible to decompose a multivariate random quantity of order n inside of a linear space provided with a quadratic metric in order to compute further summary indices
If we decompose a multivariate random quantity of order n inside of a linear space provided with a quadratic metric we observe that it is not possible to consider more than two random quantities at a time
Summary
We propose a mathematical model where the probability of an event has a concrete image [1]. The difference between two opposite points of view is well known: some scholars interpret probability as a subjective measure of the degree of belief, whereas others consider it as an objective measure connected with measurable sets [2]. We will refer to those situations characterizing economic science, statistics and other related fields of interest in which such a difference has no reason to exist because it is evident that an event cannot naturally be meant as a measurable set [3]. We have elsewhere shown that the subjective approach to decisions under uncertainty, as we propose it, has innovative contributions to offer because the probability is basically viewed as the solution to a specific decision problem rather than an opening assumption [4]
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