Page 213 - Elements of Distribution Theory

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7

Distribution Theory for Functions

of Random Variables

7.1 Introduction
Acommonprobleminstatisticsisthefollowing.Wearegivenrandomvariables X 1 ,..., X n ,
such that the joint distribution of (X 1 ..., X n )is known, and we are interested in determining
the distribution of g(X 1 ,..., X n ), where g is a known function. For instance, X 1 ,..., X n
might follow a parametric model with parameter θ and g(X 1 ,..., X n ) might be an estimator
or a test statistic used for inference about θ.In order to develop procedures for inference
about θ we may need certain characteristics of the distribution of g(X 1 ,..., X n ).

Example 7.1 (Estimator for a beta distribution). Let X 1 ,..., X n denote independent,
identically distributed random variables, each with an absolutely continuous distribution
with density

θx θ−1 , 0 < x < 1
where θ> 0. This is a special case of a beta distribution.
Consider the statistic
n
1
log X j ;
n
j=1
this statistic arises as an estimator of the parameter θ.In carrying out a statistical analy-
sis of this model, we may need to know certain characteristics of the distribution of this
estimator.

In the earlier chapters, problems of this type have been considered for specific examples;
in this chapter we consider methods that can be applied more generally.

7.2 Functions of a Real-Valued Random Variable

First consider the case in which X is a real-valued random variable with a known distribution
and we want to determine the distribution of Y = g(X) where g is a known function. In
principle, this is a straightforward problem. Let X denote the range of X and let Y = g(X)
denote the range of Y so that g : X → Y.For any subset A of Y,

Pr(Y ∈ A) = Pr(X ∈{x ∈ X: g(x) ∈ A}) = dF X (x), (7.1)
{x∈X: g(x)∈A}
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