Fundamentals of Stochastic Networks

1

BASIC CONCEPTS IN PROBABILITY

1.1 INTRODUCTION

The concepts of experiments and events are very important in the study of probability. In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. When we perform a random experiment, the collection of possible elementary outcomes is called the sample space of the experiment, which is usually denoted by Ω. We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed. The elementary outcomes of an experiment are called the sample points of the sample space and are denoted by w_i, i = 1, 2, … If there are n possible outcomes of an experiment, then the sample space is Ω = {w₁, w₂, … , w_n}. An event is the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment. Thus, an event is a subset of the sample space.

1.2 RANDOM VARIABLES

Consider a random experiment with sample space Ω. Let w be a sample point in Ω. We are interested in assigning a real number to each w ∈ Ω. A random variable, X(w), is a single-valued real function that assigns a real number, called the value of X(w), to each sample point w ∈ Ω. That is, it is a mapping of the sample space onto the real line.

Generally a random variable is represented by a single letter X instead of the function X(w). Therefore, in the remainder of the book we use X to denote a random ...