Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric

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.

Summarized distributions of mass: a statistical approach to consumers’ consumption spaces

“…where the determinant of the square matrix of order 2 under consideration is a bilinear function of the columns of it. We note that it turns out to be (1) t, (2) t α = (2) t, (1)…”

“…. , y m inside of a linear space over R (see also [2] with respect to distributions of mass studied inside of a linear space furnished with a measure). Each proposition under consideration can be subjected to infinite translations, so it is also possible to write {y 1 + λ, y 2 + λ, .…”

“…Let 1 Y and 2 Y be two random quantities, where the number of the possible values for each of them is equal to m. (2) y m } be the sets of possible values for 1 Y and 2 Y . The two vectors identifying the possible values for 1 Y and 2 Y are linearly independent.…”

“…We consequently note that it always depends on the angle between (1) t and (2) t, with (1) t, (2) t ∈ E m . An appropriate numerical example verifying all of this requires some mathematical steps.…”

The consumer’s demand functions defined to study contingent consumption plans

“…It follows that each consumer is faced with m masses denoted by p 1 , p 2 , … , p m such that it is possible to write p 1 + p 2 + … + p m = 1 . They are located on m real numbers denoted by x 1 , x 2 , … , x m (see also Angelini and Maturo (2020)).…”