On the New Notion of the Set-Expectation for a Random Set of Events

The paper introduces new notion for the set-valued mean set of a random set. The means are defined as families of sets that minimize mean distances to the random set. The distances are determined by metrics in spaces of sets or by suitable generalizations. Some examples illustrate the use of the new definitions.mean random set, metrics in set space, mean distance, Aumann expectation, Frechet expectation, Hausdorff metric, random finite set, mean set, set-median, set-expectation

Postulating the theory of experience and chance as a theory of co~events (co~beings)

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author’s main idea was the natural inclusion of Kolmogorov’s axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The analogy between the measure of a set and the probability of an event has become clear for a long time. This analogy also allows further evolution: the measure of a set is completely analogous to the believability of an event. In order to postulate the theory of experience and chance on the basis of this analogy, you just need to add to the Kolmogorov probability theory its dual reflection — the believability theory, so that the theory of experience and chance could be postulated as the certainty (believability-probability) theory on the Cartesian product of the probability and believability spaces, and the central concept of the theory is the new notion of co~event as a measurable binary relation on the Cartesian product of sets of elementary incomes and elementary outcomes. Attempts to build the foundations of the theory of experience and chance from this general point of view are unknown to me,\ud and the whole range of ideas presented here has not yet acquired popularity even in a narrow circle of specialists; in addition, there was still no complete system of the postulates of the theory of experience and chance free from unnecessary complications. Postulating the theory of experience and chance can be carried out in different ways, both in the choice of axioms and in the choice of basic concepts and relations. If one tries to achieve the possible simplicity of both the system of axioms and the theory constructed from it, then it is hardly possible to suggest anything other than axiomatization of concepts co~event and its certainty (believability-probability). The main result of this work is the axiom of co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work, a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment

Eventologically multivariate extensions of probability theory’s limit theorems

Eventologically multivariate extensions of probability theory’s limit theorems are proposed. Eventologically multivariate version of limit theorems extends its classical probabilistic interpretation and involves into its structure of dependencies of arbitrary set of events which appears in sequence of independent tests.Event, probability, set of events, Bernoulli univariate test, Bernoulli multivariate test, eventological distribution, multivariate discrete distribution, limit theorem.

Eventology of random-fuzzy events

A brief introduction to the eventology, which has originated recently as a new line of probability theory. This line studies eventological motion of random-fuzzy events (eventological motion of events motion of matter or motion of mind changing the eventological distributions), introduces the mind as an eventological distribution to scientific and mathematical research, and absorbs the famous theory of fuzzy sets by Zadeh as a very particular approach, that still has not been grounded strictly from eventological point of view.event, probability, fuzzy event, eventology

Eventological Theory of Decision-Making

The eventological theory of decision-making, the theory of event-based decision-making is a theory of decision-making based on eventological principles and using results of mathematical eventology [1]; a theoretical basis of the practical eventology [2, 3, 4]. The beginnings of this theory which have arisen from event-based representation of the reasonable subject and his decisions in the form of eventological distributions (E-distributions) of sets of events [5] and which are based on the eventological H-theorem [6] are offered. The illustrative example of the eventological decision-making by the reasonable subject on his own event-based behaviour in the financial or share market is considered.eventology, event-based decision-making, eventological H-theorem