Probabilities

TL;DR

This book introduces probability theory fundamentals at bachelor level, covering basic concepts and advanced topics like measure theory, stochastic processes, and convergence, with exercises and solutions.

Contribution

Provides an accessible English translation of a comprehensive probability textbook, including both introductory and advanced topics with extensive exercises.

Findings

Includes detailed exercises with solutions in every chapter.

Covers a wide range of topics from basic probability to stochastic processes.

Designed for self-study at bachelor and master levels.

Abstract

Probabilities is the English translation of the book Probabilit\'es Tome 1 and Tome 2. The mathematic content is authored by Prof. Jean-Yves Ouvrard. The English version has been done by his eldest son Dr. Xavier Ouvrard. In this first version, only the first part is released. Part 1 contains 7 chapters and corresponds to bachelor level. The first part introduces the fundamentals of probability theory across 7 chapters, targeting bachelor level, including event algebras, random variables, independence, conditional probabilities, moments of discrete and continuous random variables, generating functions, and limit theorems. The second part contains 10 chapters and corresponds to master level. Following a brief introduction to measure theory, this part develops more advanced topics: probability measures and their complements, distributions and moments of random variables, modes of convergence, laws of large numbers, conditional expectation, Fourier transforms and characteristic functions, Gaussian random variables, convergence of measures, convergence in distribution, discrete-time stochastic processes, martingales, and Markov chains. The reader's work is greatly facilitated by the inclusion, in every chapter, of numerous exercises, all accompanied by detailed solutions that often provide substantial extensions to the theoretical material. Any feedback is welcome, at probabilities@xerox.mozmail.com The content is released in CC-BY-NC-SA.