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Introduction; Review of Random Variables

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Probability   ::   Set Theory   ::   Definition   ::   Conditional   ::   Random   ::   Distribution   ::   R.V.S.

Definition of probability

We now formally define what we mean by a probability space. A probability space has three components.

The first is the sample space , which is the collection of all possible outcomes of some experiment. The outcome space is frequently denoted by .

We deal with subsets of . For example, we might have an event which is "all even throws of the die'' or "all outcomes ''. We denote the collection of subsets of interest as . The elements of (that is, the subsets of ) are called events . is called the event class . We will restrict to be a -field .

The tuple is called a pre-probability space (because we haven't assigned probabilities yet). This brings us to the third element of a probability space: Given a pre-probability space , a probability distribution or a measure on is a mapping from to (which we will write: ) with the properties:

• (this is a normalization that is always applied for probabilities, but there are other measures which don't use this)
• . (Measures are nonnegative)
• If such that for all (that is, the sets are "disjoint'' or "mutually exclusive'') then

These three properties are called the axioms of probability .

The triple is called a probability space .

• tells what individual outcomes are possible
• tells what sets of outcomes -- events -- are possible.
• tells what the probabilities of these events are.

Some properties of probabilities (which follow from the axioms):

1. .

2. .

3. If then

There is also the continuity of probability property. Suppose (this is called an increasing sequence). Define

We write this as ( converges up to ). Similarly, if

(a decreasing sequence), define

If , then . If , then .

We will introduce more properties later.

Some examples of probability spaces

1. (a discrete set of outcomes). . Let be a set of nonnegative numbers satisfying . Define the function by

Then is a probability space.
2. Uniform distribution: Let , =smallest -field containing all intervals in [0,1]. We can take (without proving that this actually works -- but it does)

for , and

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Introduction; Review of Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec1_3.html. This work is licensed under a Creative Commons License