Introduction to Mathematical Probability Theory

1971 ◽  
Vol 39 (1) ◽  
pp. 108
Author(s):  
J. Gani ◽  
M. Eisen
Keyword(s):  
Probability Theory ◽  
Mathematical Probability

scholarly journals The Maxim of Probabilism, with special regard to Reichenbach

Synthese ◽  
2021 ◽  
Author(s):  
Miklós Rédei ◽  
Zalán Gyenis
Keyword(s):  
Probability Theory ◽  
Probability Measure ◽  
Measure Theory ◽  
Necessary Condition ◽  
Special Regard ◽  
Foundations Of Probability ◽  
Mathematical Probability ◽  
Conditional Expectations ◽  
Measure Spaces

AbstractIt is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations.

scholarly journals The Paradigm of Complex Probability and Thomas Bayes’ Theorem

2021 ◽  
Author(s):  
Abdo Abou Jaoude
Keyword(s):  
Probability Theory ◽  
Bayes Theorem ◽  
Stochastic Variable ◽  
The Real ◽  
Mathematical Probability ◽  
Variable Distribution ◽  
Probability Concept

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to gauge in the sets R, M, and C all the corresponding probabilities. Hence, the probability in the entire set C = R + M is incessantly equal to one independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be evaluated totally in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. We will apply this novel paradigm to the classical Bayes’ theorem in probability theory.

scholarly journals PHILOSOPHICAL PROBLEMS OF THE PROBABILITY THEORY

2020 ◽  
pp. 31-36
Author(s):  
Татьяна Геннадьевна Стоцкая
Keyword(s):  
Probability Theory ◽  
Human Life ◽  
Philosophical Approach ◽  
Mathematical Probability ◽  
Basic Concepts ◽  
Definition Of ◽  
Fundamental Categories

В настоящее время для решения многих практических задач в различных сферах человеческой жизни мы используем разные математико-статические методы. Данные методы базируются на основных понятиях и положениях теории вероятности. Проблема теории вероятности кроется в самой природе вероятности, раскрыть которую способен исключительно философский подход. Традиционно принято считать, что именно математическая теория вероятностей формулирует самое строгое определение вероятности. Но применение данного подхода гораздо шире: понятие вероятности стало одной из фундаментальных категорий биологии, космологии, кибернетики, физики. Currently, to solve many practical problems in various fields of human life, we use different mathematical and static methods, these methods are based on the basic concepts and statements of probability theory. The problem of probability theory lies in the very nature of probability, which can be revealed by a philosophical approach. Traditionally, it is considered that mathematical probability theory formulates the strictest definition of probability. But the application of this approach is much broader: the concept of probability has become one of the fundamental categories of biology, cosmology, Cybernetics, and physics.

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