## Probability and Stochastic Processes

Introduction

What is Probability? In essence:

Mathematical modeling of random events and phenomena. It is fundamentally different from modeling deterministic events and functions, which constitutes the traditional study of Mathematics.

However, the study of probability uses concepts and notions straight from Mathematics; in fact Measure Theory and Potential Theory are expressions of abstract mathematics generalizing the Theory of Probability.

Like so many other branches of mathematics, the development of probability theory has been stimulated by the variety of its applications. In turn, each advance in the theory has enlarged the scope of its influence. Mathematical statistics is one important branch of applied probability; other applications occur in such widely different fields as genetics, biology, psychology, economics, finance, engineering, mechanics, optics, thermodynamics, quantum mechanics, computer vision, geophysics,etc. In fact I compel the reader to find one area in today's science where no applications of probability theory can be found.

Early history

In the XVII-th century the first notions of Probability Theory appeared. More precisely, in 1654 Antoine Gombaud Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, was puzzled by an apparent contradiction concerning a popular dice game. The game consisted of throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable (based on the payoff of the game). However, his own calculations based on many repetitions of the 24 throws indicated just the opposite. Using modern probability language de Méré was trying to establish if such an event has probability greater than 0.5 (we are looking at this question in example 1.7). Puzzled by this and other similar gambling problems he called on the famous mathematician BlaisePascal. This, in turn led to an exchange of letters between Pascal and another famous French mathematician Pierre de Fermat. This is the first known documentation of the fundamental principles of the theory of probability. Before this famous exchange of letters, a few other simple problems on games of chance had been solved in the XV-th and XVI-th centuries by Italian mathematicians; however, no general principles had been formulated before this famous correspondence.

In 1655 during his first visit to Paris, the Dutch scientist Christian Huygens learned of the work on probability carried out in this correspondence. On his return to Holland in 1657, Huygens wrote a small work De Ratiociniis in Ludo Aleae , the first printed work on the calculus of probabilities. It was a treatise on problems associated with gambling. Because of the inherent appeal of games of chance, probability theory soon became popular, and the subject developed rapidly during the XVIII-th century.

The XVIII-th century

The major contributors during this period were Jacob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754). Jacob (Jacques) Bernoulli was a Swiss mathematician who was the first to use the term integral. He was the first mathematician in the Bernoulli family, a family of famous scientists of the XVIII-th century. Jacob Bernoulli's most original work was Ars Conjectandi published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it still was a work of the greatest significance in the development of the Theory of Probability. De Moivre was a French mathematician who lived most of his life in England 1 . De Moivre pioneered the modern approach to the Theory of Probability, in his work The Doctrine of Chance: A Method of Calculating the Probabilities of Events i