Page 23 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

2 1 Introduction

In the case of the body fall there is a law that allows the exact computation of
one of the variables h or t (for given h 0 and g) as a function of the other one.
Moreover, if we repeat the body-fall experiment under identical conditions, we
consistently obtain the same results, within the precision of the measurements.
These are the attributes of deterministic data: the same data will be obtained,
within the precision of the measurements, under repeated experiments in well-
defined conditions.
Imagine now that we were dealing with Stock Exchange data, such as, for
instance, the daily share value throughout one year of a given company. For such
data there is no known law to describe how the share value evolves along the year.
Furthermore, the possibility of experiment repetition with identical results does not
apply here. We are, thus, in presence of what is called random data.
Classical examples of random data are:

− Thermal noise generated in electrical resistances, antennae, etc.;
− Brownian motion of tiny particles in a fluid;
− Weather variables;
− Financial variables such as Stock Exchange share values;
− Gambling game outcomes (dice, cards, roulette, etc.);
− Conscript height at military inspection.

In none of these examples can a precise mathematical law describe the data.
Also, there is no possibility of obtaining the same data in repeated experiments,
performed under similar conditions. This is mainly due to the fact that several
unforeseeable or immeasurable causes play a role in the generation of such data.
For instance, in the case of the Brownian motion, we find that, after a certain time,
the trajectories followed by several particles that have departed from exactly the
same point, are completely different among them. Moreover it is found that such
differences largely exceed the precision of the measurements.
When dealing with a random dataset, especially if it relates to the temporal
evolution of some variable, it is often convenient to consider such dataset as one
realization (or one instance) of a set (or ensemble) consisting of a possibly infinite
number of realizations of a generating process. This is the so-called random
process (or stochastic process, from the Greek “stochastikos” = method or
phenomenon composed of random parts). Thus:

− The wandering voltage signal one can measure in an open electrical
resistance is an instance of a thermal noise process (with an ensemble of
infinitely many continuous signals);
− The succession of face values when tossing n times a die is an instance of a
die tossing process (with an ensemble of finitely many discrete sequences).
− The trajectory of a tiny particle in a fluid is an instance of a Brownian
process (with an ensemble of infinitely many continuous trajectories);