Page 25 - Computational Statistics Handbook with MATLAB

Chapter 2

Probability Concepts

2.1 Introduction
A review of probability is covered here at the outset because it provides the
foundation for what is to follow: computational statistics. Readers who
understand probability concepts may safely skip over this chapter.
Probability is the mechanism by which we can manage the uncertainty that
underlies all real world data and phenomena. It enables us to gauge our
degree of belief and to quantify the lack of certitude that is inherent in the
process that generates the data we are analyzing. For example:

• To understand and use statistical hypothesis testing, one needs
knowledge of the sampling distribution of the test statistic.
• To evaluate the performance (e.g., standard error, bias, etc.) of an
estimate, we must know its sampling distribution.
• To adequately simulate a real system, one needs to understand the
probability distributions that correctly model the underlying pro-
cesses.
• To build classifiers to predict what group an object belongs to based
on a set of features, one can estimate the probability density func-
tion that describes the individual classes.

In this chapter, we provide a brief overview of probability concepts and
distributions as they pertain to computational statistics. In Section 2.2, we
define probability and discuss some of its properties. In Section 2.3, we cover
conditional probability, independence and Bayes’ Theorem. Expectations are
defined in Section 2.4, and common distributions and their uses in modeling
physical phenomena are discussed in Section 2.5. In Section 2.6, we summa-
rize some MATLAB functions that implement the ideas from Chapter 2.
Finally, in Section 2.7 we provide additional resources for the reader who
requires a more theoretical treatment of probability.

© 2002 by Chapman & Hall/CRC