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Chapter 2

Mathematical Statistics

In this chapter, we turn to the study of statistics, which is concerned with the analysis of
experimental data. In a book of this nature we cannot hope to do justice to such a large subject;
indeed, many would argue that statistics belongs to the realm of experimental science rather
than in a mathematics textbook. Nevertheless, physical scientists and engineers are regularly
called upon to perform a statistical analysis of their data and to present their results in a
statistical context. Therefore, we will concentrate on this aspect of a much more extensive
subject.

2.1. Experiments, samples and populations

We may regard the product of any experiment as a set of N measurements of some quantity x
or set of quantities x, y, . . . , z. This set of measurements constitutes the data. Each measurement
(or data item) consists accordingly of a single number x i or a set of numbers (x i , y i , . . . , z i ), where
i = 1, . . . , N. For the moment, we will assume that each data item is a single number, although
our discussion can be extended to the more general case.
As a result of inaccuracies in the measurement process, or because of intrinsic variability in
the quantity x being measured, one would expect the N measured values x 1 , x 2 , . . . , x N to be
different each time the experiment is performed. We may therefore consider the x i as a set of
N random variables. In the most general case, these random variables will be described by some
N-dimensional joint probability density function P(x 1 , x 2 , . . . , x N ). In other words, an experiment
consisting of N measurements is considered as a single randomsample from the joint distribution
(orpopulation)P(x),wherexdenotesapointintheN-dimensionaldataspacehavingcoordinates
(x 1 , x 2 , . . . , x N ).
The situation is simplified considerably if the sample values x i are independent. In this case,
the N-dimensional joint distribution P(x) factorizes into the product of N one-dimensional
distributions,
P(x) = P(x 1 )P(x 2 ) · · · P(x N ). (1.1)

In the general case, each of the one-dimensional distributions P(x i ) may be different. A typical
example of this occurs when N independent measurements are made of some quantity x but the
accuracy of the measuring procedure varies between measurements.
It is often the case, however, that each sample value x i is drawn independently from the same
population. In this case, P(x) is of the form (1.1), but, in addition, P(x i ) has the same form for
each value of i. The measurements x 1 , x 2 , . . . , x N are then said to form a random sample of size
N from the one-dimensional population P(x). This is the most common situation met in practice
and, unless stated otherwise, we will assume from now on that this is the case.

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