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 ...