2 - Stochastic variables and processes

Abstract

This chapter reviews elements of random variables and fundamentals of stochastic processes. A random variable (stochastic variable) is defined by a set (Ω) of possible values and a probability distribution (P) over this set. The chapter focuses on the real random variables for which Ω is a subset of the real numbers and with collections of real random variables indexed by vector-valued random variables or by time and/or space. There are two types of real random variables: (1) continuous random variables—whose set of possible values is an interval on the real line and (2) discrete random variables—whose sample space consists of a finite or countable infinite subset of real numbers. The chapter also introduces several mathematical tools, such as Dirac delta function, Fourier transform, and Laplace transform that are frequently used f to study flow in porous media.