Bayesian Statistics: An Introduction, 4th Edition

3.5 The uniform distribution

3.5.1 Preliminary definitions

The support of a density is defined as the set of values of x for which it is non-zero. A simple example of a family of densities in which the support depends on the unknown parameter is the family of uniform distributions (defined later). While problems involving the uniform distribution do not arise all that often in practice, it is worth while seeing what complications can arise in cases where the support does depend on the unknown parameter.

It is useful to begin with a few definitions. The indicator function of any set A is defined by

This is sometimes called the characteristic function of the set A in some other branches of mathematics, but not in probability and statistics (where the term characteristic function is applied to the Fourier–Stieltjes transform of the distribution function).

We say that y has a Pareto distribution with parameters and γ and write

if it has density

This distribution is often used as a model ...