Preface

Probability theory is a branch of statistics that employs mathematical methods of collection, organization, and interpretation of data, with applications in practically all scientific areas. When working with probability theory, we analyze random phenomena and assess the likelihood that an event will occur. This book, *Applied Probability Theory - New Perspectives, Recent Advances and Trends*, discusses some fundamental aspects of probability theory and explores its use to solve a large array of problems. Chapters address such topics as complex probability, the stability of algorithms in statistical modeling, the non-homogeneous Hofmann process, and more.

Each time I work in the field of mathematical probability and statistics I find pleasure in tackling the knowledge, theorems, proofs, and applications of the theory. Each problem is a riddle to be solved and I become relieved and extremely happy when I reach the riddle's solution. This proves two important facts: first, the power of mathematics and its models to deal with such kinds of problems and second, the power of the human mind to understand such problems and tame the wild concepts of randomness, probability, stochasticity, uncertainty, chaos, chance, and non-determinism.

I chose the word paradigm for this branch of mathematical sciences after consulting the influential book *The Structure of Scientific Revolutions* by Thomas Kuhn, in which the author used the term to describe a set of theories, standards, and methods that together represent a way of organizing knowledge, that is, a model or a way of viewing the world. Kuhn stated in his thesis that revolutions in science occur when an older paradigm is reexamined, rejected, and replaced by another, just like Einstein's theories of special and general relativity that dethroned Newtonian mechanistic theory, or quantum mechanics that replaced the classical theories of electromagnetism and thermodynamics when probing the micro-world. What about probability and statistics? We can affirm that their set of theories and methods developed across the centuries have defined for us a way to view the world and a model to understand and deal with such concepts as randomness, chance, stochasticity, chaos, probability, and so on. Hence, the definition of a paradigm suits very well this discipline of knowledge and this methodology of thinking. This justifies my usage of this term in my two chapters of this book.

I hope that after reading this book you will recognize my amazement and wonder at the power of the theory of probability and statistics to deal with randomness, as well as my excitement to delve into the depths of a very profound field in mathematics. Thus, to convey my impression of wonder I cite the following words of Albert Einstein:

"The most incomprehensible thing about the universe is that it is comprehensible…"

Furthermore, 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."

Some may criticize my opinion and say that the theory of probability and statistics is a speculative and an uncertain science dealing with approximations and uncertainties. That is completely true. But since this field, or paradigm, is a part of mathematics, it has allowed us to understand, measure quantitatively, and tame chaos, even if not completely and absolutely. In fact, probability theory keeps the spirit and the flavor of "exact" sciences through its numbers, proofs, figures, theorems, and graphs.

To conclude, I am truly astonished by the power of probability theory to deal with random data and phenomena, and this feeling and impression have never left me since the first time I was introduced to this branch of science and mathematics. I hope that this book will convey and share this feeling with readers.

> **Abdo Abou Jaoudé, Ph.D.** Notre Dame University-Louaizé, Zouk Mosbeh, Lebanon

**Chapter 1**

Function

**Abstract**

*Abdo Abou Jaoudé*

The Paradigm of Complex

Probability and Quantum

completely deterministically in the universe **C** ¼ **R** þ**M**.

*lack of knowledge."*

**Keywords:** chaotic factor, degree of our knowledge, complex random vector,

*"You believe in the God who plays dice, and I in complete law and order."*

*"Without mathematics, we cannot penetrate deeply into philosophy. Without philosophy, we cannot penetrate deeply into mathematics. Without both, we cannot penetrate deeply into anything … "*.

*"Nothing in nature is by chance … Something appears to be chance only because of our*

*Baruch Spinoza.*

*Albert Einstein, Letter to Max Born*

*Gottfried Wilhelm von Leibniz.*

probability norm, complex probability set **C**, position wave function

Mechanics: The Infinite Potential

Well Problem – The Position Wave

The system of axioms for probability theory laid in 1933 by 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 set **C** 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**. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in **C**. My innovative complex probability paradigm (*CPP*) will be applied to the established theory of quantum mechanics in order to express it
