What is it about?
Probability theory is a branch of statistics, a science that employs mathematical methods of collection, organization, and interpretation of data, with applications in practically all scientific areas. This book provides a comprehensive overview of probability theory. It discusses some fundamental aspects of pure and applied probability theory and explores its use in solving a large array of problems. Topics addressed include complex probability, the stability of algorithms in statistical modeling, the non-homogeneous Hofmann process, and more.
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Why is it important?
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…”
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: Applied Probability Theory - New Perspectives, Recent Advances and Trends, January 2023, IntechOpen,
DOI: 10.5772/intechopen.104026.
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