2
Probability
Distributions
In Chapter 1, we emphasized the central role played by probability theory in the
solution of pattern recognition problems. We turn now to an exploration of some
particular examples of probability distributions and their properties. As well as be-
ing of great interest in their own right, these distributions can form building blocks
for more complex models and will be used extensively throughout the book. The
distributions introduced in this chapter will also serve another important purpose,
namely to provide us with the opportunity to discuss some key statistical concepts,
such as Bayesian inference, in the context of simple models before we encounter
them in more complex situations in later chapters.
One role for the distributions discussed in this chapter is to model the prob-
ability distributionp(x)of a random variablex, given a finite setx 1 ,...,xNof
observations. This problem is known asdensity estimation. For the purposes of
this chapter, we shall assume that the data points are independent and identically
distributed. It should be emphasized that the problem of density estimation is fun-