THE XVII CONFERENCE ON FAMEMS AND THE III WORKSHOP ON HILBERT’S SIXTH PROBLEM,KRASNOYARSK,SIBERIA,RUSSIA, 2018 Experienced, random, and experienced-random variables with their distribution functions in the theory of co∼events Oleg Yu. Vorobyev Institute of mathematics and computer science Siberian Federal University Krasnoyarsk mailto:oleg.yu.vorobyev@gmail.com http://www.sfu-kras.academia.edu/OlegVorobyev http://olegvorobyev.academia.edu Abstract: Definitions of the distribution functions of experienced, random, and experienced-random variables are given. The familiar apparatus of distribution functions is used here to characterize three types of variables that naturally arise in the co∼event mechanics and among which two types, experienced and experienced- random variables, are completely new concepts. The first type, the experienced variable, is the numerical function of past experience and past causes. The second type, a random variable, is a numerical function of a future case and a future consequence. Finally, the third type, experienced-random variable, is a numerical function that relates the past experience of the observer with the future case of observation, i.e. past cause with future consequence. These definitions are illustrated by a large number of examples that have a transparent interpretation in numerous applications. Keywords: Eventology, event, probability theory, Kolmogorov’s axiomatics, co∼event, bra-event, ket-event, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼eventum mechanics, experienced variable, random variable, experienced-random variable, distribution function, reflexion, expectation, existence. MSC: 60A05, 60A10, 60A86, 62A01, 62A86, 62H10, 62H11, 62H12, 68T01, 68T27, 81P05, 81P10, 91B08, 91B10, 91B12, 91B14, 91B30, 91B42, 91B80, 93B07, 94D05 I will remind you that the theory of experience and chance is based on the dual axiomatics of co∼events, which includes Kolmogorov’s axiomatics of probability theory as one of the dual halves [1, 2, 3]. Defined by this axiomatics the duality of a co∼event leads to the duality of any numerical superstructure of this new theory, which also, as its co∼event foundation, includes the classical concept of a random variable (r.v.) as a dual reflection of the new concept of an experienced variable (e.v.), and introduces a fundamentally new concept of experienced-random variable (e-r.v.) defined on the Cartesian product ⟨Ω|Ω⟩ = ⟨Ω|×|Ω⟩ of the space of elementary experienced (accumulated) incomes ⟨Ω| and the space of elementary random outcomes |Ω⟩. 1 Experienced, random and experienced-random variables Experienced, random and experienced-random variables are a part of the basic concepts of the theory of experience and chance. Complete and free from any unnecessary restrictions the presentation of the foundations of the theory of probabilities on the basis of measure theory is given by Kolmogorov [4, 1933]; it made it quite obvious that the random variable is nothing more than a measurable function on the probability space. The theory of experience and chance also relies on the measure theory, which makes it equally obvious that the experienced variable dual to random one, in turn, is nothing more than a measurable function on the believability space dual to the probability one. An experienced-random variable is defined as a measurable function on the Cartesian product of believability and probability spaces, the certainty space. These circumstances can not be ignored in the presentation of the beginning of the theory of experience and chance, which succeeded in combining the theory of believabilities and the theory of probabilities on the basis of the concepts of the space of elementary incomes and the space of elementatry outcomes and their Cartesian product, the space of elementary incomes-outsomes, and one must not forget, each time emphasizing, that only when one is immersed in a dual context of the theory of experience and chance, representations about experienced, random and experienced-random variables acquire the mathematical and applied content. c ○ 2018 O.Yu.Vorobyev This is an open-access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited. Oleg Vorobyev (ed.), Proc. of the XVII FAMEMS’2018, Krasnoyarsk: SFU, ISBN 978-5-9903358-8-2