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Thus, if the same quantity is measured several times in conditions that are as-
sumed to be identical, the results of the measurements are not identical. Black-
box modeling aims at finding, from available measurements, a mathematical
expression that provides an estimate of what the result of the measurement
would be in the absence of disturbances, or, in other words, at finding a de-
terministic relation, if any, between the factors
x
and the quantity of interest
y
p
. Statistics provide the conceptual framework that is suitable for that task.
Therefore, the chapter starts with the introduction of elementary vocabulary
and concepts of statistics; some examples are developed in the additional ma-
terial at the end of the chapter. The reader who has some familiarity with
statistics may skip the next section.
2.2 Elementary Concepts and Vocabulary of Statistics
There are many classical textbooks in statistics (see or instance [Mood et al.
1974; Wonnacott et al. 1990]) to which the reader can refer for many more
details and for the proofs of some results.
2.2.1 What is a Random Variable?
A random variable is a very convenient mathematical concept for dealing with
quantities (such as results of measurements) whose values are uncertain: their
values is considered as a realization of a random variable. The latter is fully
defined by its probability density or distribution.
Denoting by
p
Y
(
y
)the
probability distribution
function (pdf) of a random
variable
Y
, the probability that the value of a realization of
Y
lie between y
and
y
+
dy
is equal to
p
Y
(
y
)
dy
.
Thus, modeling a measurable quantity
y
p
by a random variable
Y
is equiv-
alent to assuming that the result of a measurement is the result of a random
choice of a value
y
with a (generally unknown) probability distribution
p
Y
(
y
).
Modeling a quantity of interest by a random variable is definitely not equiv-
alent to stating or assuming that the quantity of interest is not governed by
a deterministic process: it is just a convenient mathematical trick for dealing
with the fact that some factors that have an influence on the result of the
measurement are not known, or are known but neither measured nor con-
trolled (maybe because they are neither measurable nor controllable, such as
wind gusts in the modeling of airplane flight).
Property.
The probability distribution function is the derivative of the cumu-
lative distribution function:. p
Y
(
y
)=(
dF
Y
(
y
))
/
(
dy
)with
F
Y
(
y
)=
Probability
(
Y
≤
y
)
Because any realization
y
of the random variable
Y
lies between
−∞
and
, one has:
+
∞
−∞
+
∞
p
Y
(y)
dy
=1.
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