Foundations of Probability Theory

Abstract

This chapter introduces the fundamental notions of probability theory: random variables, law, expected value, variance, moments of random variables, characteristic functions, etc. We list several classical probability laws, some of which will play a major role in the forthcoming chapters. We also restate some basic notions and facts of measure theory in the language of probability theory, and we apply the Hilbert space structure of L 2 to the linear regression problem, which consists in finding the best approximation of a (real) random variable as an affine function of several other random variables. The end of the chapter gives several ways of characterizing the law of a random variable with values in or in