Understanding the Sample Space in Probability Theory
Summary
In this lesson, we explore the concept of Probability Space, a mathematical structure consisting of a Sample Space, Sigma-Algebra, and Probability Measure. We examine the Sample Space in detail, understood as the collection of all possible states of a random process. Through practical examples, we illustrate the construction of discrete and continuous sample spaces, and explain how measurable events are constructed from them and how probability measures are calculated. This lesson is fundamental to understanding the foundations of Probability Theory and lays the groundwork for its application in various fields.
LEARNING OBJECTIVES:
Upon completing this lesson, students will be able to:
- Understand the concept of Probability Space.
- Identify the elements that make up the Probability Space.
- Differentiate between discrete and continuous sample spaces.
- Construct discrete and continuous sample spaces.
TABLE OF CONTENTS
THE PROBABILITY SPACE
EXAMPLES OF SAMPLE SPACES
DISCRETE AND CONTINUOUS SAMPLE SPACES
The Probability Space
Probability Theory is founded on an object called Probability Space. This is a mathematical structure composed of: (i) a Sample Space \Omega, (ii) a Sigma-Algebra \Sigma, and (iii) a Probability Measure P. To construct the probability space, we first review the concept of sample space.
The union of all possible states \omega of a random process forms a non-empty set \Omega which we call Sample Space.
Examples of Sample Spaces
| EXAMPLE 1 |
| If we toss a coin, then we have two possible outcomes: Heads (H) and Tails (T). Therefore, the sample space will be \Omega_{1m}=\{H,T\} |
| EXAMPLE 2 |
| If the previous experiment is repeated, but now with two tosses, then we will have: \Omega_{2m}=\{(H,H);(H,T);(T,H);(T,T)\} That is, all possible ways of arranging heads and tails in groups of two. |
| EXAMPLE 3 |
| Rolling a six-faced die has the following sample space: \Omega_{1d6}=\{1,2,3,4,5,6\} That is, the number indicated on each of its faces. |
| EXAMPLE 4 |
| The lifespan of an electrical appliance (measured in hours) has a sample space of the form \Omega_{ae}=\{t\in \mathbb{R} \;|\; t\geq 0\} That is, the lifespan of the appliance is a number t contained in the interval [0,+\infty[ |
Discrete and Continuous Sample Spaces
From these examples, we can distinguish between two types of sample spaces, these are discrete and continuous. Discrete sample spaces, as in the first three examples, consist of finite sets, although they can also be infinite and countable (like any subset of \mathbb{N}). On the other hand, continuous sample spaces are infinite and uncountable sets; they are generally represented by sub-intervals of \mathbb{R}.
From the elements of the sample space (the possible states), measurable events (objects of the sigma-algebra) of the probability space are constructed, and it is on these objects that probability measures are calculated.