# The Paradigm of Complex Probability, the Law of Large Numbers, and the Central Limit Theorem

Abdo Abou Jaoudé
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon.

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### Book Details

Author(s) Abdo Abou Jaoudé 164 B P International English 978-81-970571-3-7 (Print) 978-81-970571-4-4 (eBook) February 21, 2024

### About The Author / Editor

#### Abdo Abou Jaoudé

Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon.

This book is titled “The Paradigm of Complex Probability, The Law of Large Numbers, and The Central Limit Theorem”. It embraces an inventive work that illustrates my original and new “Complex Probability Paradigm”, or CPP for short, when applied to two famous problems and theorems in Mathematics which are: The Law of Large Numbers first and The Central Limit Theorem second. Consequently, two chapters were written for this purpose where my novel paradigm was implemented to each topic.

Furthermore, to introduce and to summarize this innovative paradigm to the reader we state that: The system of axioms for probability theory laid in 1933 by the great Russian mathematician Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex probability set C = R + M instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the “real” laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that chance and luck in R is replaced by total determinism in C = R + M. Accordingly, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon, met in science and in mathematics, in C. Thus, since this extension was found to be fruitful and successful and rewarding, then a novel paradigm of stochastic sciences and prognostic and mathematics and science in general was established in which all stochastic phenomena in R were expressed entirely deterministically in the probability universe C = R + M. This is what I coined by the terms “The Complex Probability Paradigm”.

As a matter of fact, my background is a Ph.D. in Computer Science, a Ph.D. in Applied Mathematics, and a Ph.D. in Applied Probability and Statistics, so I combined my knowledge in Probability Theory with Applied Mathematics and I wrote this book and this mathematical and physical and philosophical model and paradigm. Actually, as it is well-known, the theorems and the theories of Pure and Applied Mathematics are the work of eminent mathematicians like the prodigies Carl Friedrich Gauss, Leonhard Euler, Siméon-Denis Poisson, John Von Neumann, Thomas Bayes, Jakob Bernoulli, Joseph Louis Lagrange,   Pierre-Simon Laplace, Andrey Nikolaevich Kolmogorov, Blaise Pascal,   Abraham De Moivre, etc.

Hence, we must pay tribute to those magnificent giants of science and sometimes of philosophy who contributed to enrich our knowledge of mathematics and physics and the universe and increased our understanding of all-natural phenomena and of existence.

Moreover, the book uses hence algorithms to simulate my applied paradigm to random phenomena encountered in Mathematics. Accordingly, I wrote them using the well-known software which is MATLAB version 2023 and they were executed on my workstation computer system to acquire a suitable speed and efficiency needed for this numerical and stochastic method.

Additionally, in January 2023, I have done an interview with the platform named Faculti located in London, United Kingdom, where I have discussed upon their request my published paper in 2018 with Taylor and Francis and which is: “The Paradigm of Complex Probability and Ludwig Boltzmann’s Entropy”. It lasts about 20 minutes. So, you can surely log in the website of Faculti and watch it for free.

Likewise, my pioneering paradigm was the subject of 25 international journals papers and books chapters published since 2010 till 2023 like with Taylor and Francis and with IntechOpen, London, United Kingdom.

To conclude, as I have mentioned in many previous books and publications, I repeat and say that due to its universality mathematics is the most positive and certain branch of science. Surely, the pleasure of working and doing mathematics and science is everlasting. I hope that the reader will benefit from both and share the pleasure of examining the present manuscript. Finally, I restate also my words to affirm that the combination of both mathematics and science in general leads to “magical” and amazing results and models and philosophical paradigms, and the following work is an illustration of this approach.