Notes for Math 450 Lecture Notes 2

We first explain the basic concept of a probability space, (Ω,F, P). This may be interpreted as an experiment with random outcomes. The set Ω is the collection of all possible outcomes of the experiment; F is a family of subsets of Ω called events; and P is a function that associates to an event its probability. These objects must satisfy certain logical requirements, which are detailed below. A random variable is a function X: Ω → S of the output of the random system. We explore some of the general implications of these abstract concepts. 1.1 Events and the basic set operations on them Any situation where the outcome is regarded as random will be referred to as an experiment, and the set of all possible outcomes of the experiment comprises its sample space, denoted by S or, at times, Ω. Each possible outcome of the experiment corresponds to a single element of S. For example, rolling a die is an experiment whose sample space is the finite set {1, 2, 3, 4, 5, 6}. The sample space for the experiment of tossing three (distinguishable) coins is {HHH,HHT,HTH,HTT, THH, THT, TTH, TTT