What is it about?
In the current manuscript we will apply to the well-known and established theory of Markov Chains my groundbreaking and novel Complex Probability Paradigm (or CPP) which will lead to a completely and perfectly deterministic expression of these stochastic processes in the universe of probabilities C = R + M. Consequently, it is important before “probing the depths” of Markov Chains, that we define in this chapter some essential mathematical concepts and fundamental tools and theorems that will be extensively used in the whole manuscript algorithms and C++ programs.
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Why is it important?
Calculating probabilities is the main task of the theory of classical probability. In fact, if we add new dimensions to a random phenomenon it will result to a deterministic expression of the theory of probability. This is the original and novel idea at the foundations of “The Complex Probability Paradigm (or CPP for short)”. As a matter of fact, probability theory is a nondeterministic theory in its core; that means that the outcomes of events are due to chance and randomness. If we add imaginary and new dimensions to a random experiment occurring in the set R it will result to a deterministic experiment and thus a nondeterministic phenomenon will have a certain outcome in the complex probability set C. If the random event becomes completely predictable then we will have perfect knowledge to predict the outcome of random experiments that arise in the real world in all random processes. Consequently, the work that has been accomplished in CPP was to extend the set R of real probabilities to the set C = R + M of deterministic complex probabilities by incorporating the contributions of the set M which is the imaginary probabilities set. Therefore, because this extension was found to be fruitful, then a novel paradigm of prognostic and nondeterministic sciences was established in which all random phenomena in R was defined deterministically. I called this original model "the Complex Probability Paradigm" that was initiated and illustrated in my numerous earlier research publications.
Perspectives
Although I have taught courses on probability and statistics at the university level for many years, I consider myself a beginner in this branch of knowledge; in fact an absolute beginner, always thirsty to learn and discover more. I think that the mathematician who proves to be successful in tackling and mastering the theory of probability and statistics has made it halfway to understanding the mystery of existence revealed in a universe governed sometimes in our modern theories by randomness and uncertainties. The probabilistic aspect is evident in the theories of the quantum world, of thermodynamics, or of statistical mechanics, for example. Hence, the universe’s secret code, I think, is written in a mathematical language, just as Galileo Galilei expressed it in these words: “Philosophy is written in this very great book which is the universe that always lies open before our eyes. One cannot understand this book unless one first learns to understand the language and recognize the characters in which it is written. It is written in a mathematical language and the characters are triangles, circles and other geometrical figures. Without these means it is humanly impossible to understand a word of it. Without these there is only clueless scrabbling around in a dark labyrinth.”
Dr. Abdo Abou Jaoude
Notre Dame University Louaize
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This page is a summary of: Fundamental Mathematical Methods and Concepts, August 2024, Sciencedomain International,
DOI: 10.9734/bpi/mono/978-93-48006-18-9/ch1.
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