Page 200 - Probability and Statistical Inference

4

Functions of Random Variables

and Sampling Distribution

4.1 Introduction

It is often the case when we need to determine the distribution of a function
of one or more random variables. There is, however, no one unique ap-
proach to achieve this goal. In a particular situation, one or more approaches
given in this chapter may be applicable. Which method one would ultimately
adopt in a particular situation may largely depend on one’s taste. For com-
pleteness, we include standard approaches to handle various types of prob-
lems involving transformations and sampling distributions.
In Section 4.2 we start with a technique involving distribution functions.
This approach works well in the case of discrete as well as continuous
random variables. Section 4.2 also includes distributions of order statistics
in the continuous case. Next, Section 4.3 demonstrates the usefulness of a
moment generating function (mgf) in deriving distributions. Again, this ap-
proach is shown to work well in the case of discrete as well as continuous
random variables. Section 4.4 exclusively considers continuous distribu-
tions and transformations involving one, two, and several random variables.
One highlight in this section consists of the Helmert (Orthogonal) Transfor-
mation in the case of a normal distribution, and another consists of a trans-
formation involving the spacings between the successive order statistics in
the case of an exponential distribution. In both these situations, we deal
with one-to-one transformations from n random variables to another set of
n random variables, followed by discussions on Chi-square, Student’s t,
and F distributions. Section 4.5 includes sampling distributions for both
one-sample and two-sample problems. Section 4.6 briefly touches upon the
multivariate normal distributions and provides the distribution of the Pearson
correlation coefficient in the bivariate normal case. Section 4.7 shows, by
means of several examples, the importance of independence of various ran-
dom variables involved in deriving sampling distributions such as the Student’s
t and F distributions as well as in the reproductive properties of normal and
Chi-square random variables.
The following example shows the immediate usefulness of results which
evolve via transformations and sampling distributions. We first solve the
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