Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 1

Introduction: Basic Notions of Probability Theory

 

 

1.1. Probability space

The main notion of probability theory is a probability space (Ω, , P) consisting of any set of elementary events (or outcomes) Ω, a system of events , and probability measure P. Though these objects form an unanimous whole, we shall try to consider them separately.

Sample space Ω is any non-empty set. Its elements are interpreted as all possible outcomes of an experiment (test, monitoring, phenomenon, and so on) and are called outcomes or elementary events. They are often denoted by letter ω (possibly with some index(es)). Let us consider some examples.

EXAMPLE 1A. Suppose that our experiment involves throwing a die once. Usually, we are only interested in the number of dots, and so all possible outcomes can be described by the sample space Ω = {1, 2, 3, 4, 5, 6}. Naturally, the outcome of the experiment “the number of dots that appeared on top is five” is represented by the simple event ω = 5.

EXAMPLE 1B. Consider the more complex experiment of throwing a die thrice. It can be described by the sample space consisting of all triples (i , j, k) of the natural numbers from 1 to 6 (the number of such triples is 6³ = 216):

EXAMPLE 1C. If a die is thrown an unknown (in advance) number of times (for example, ...