8
J. F. Reis et al.
Fig. 1.2 Plot of the function
in Eq. (1.6). This is the
solution of Eq. (1.5) when
k ∼ lognormal(0; 0, 1).
Space discritazation has 11
points
where u 0 and u 1 are the left and right hand-side boundary conditions. As stated
above, the solution plotted in Fig. 1.2 is obtained for the particular case k(x, θ) ≡
k ∼ lognormal(0, 0.1). In particular, the equality E[1/k] = 1/E[k] holds for such
distribution. The problem we are referring to is very simple, and there exist wellestablished procedures to compute the statistical moments of its solution. On the
other hand, if the problem was too complicated, it would be impossible to retrieve
the very same statistical information through an analytic procedure. The only way
to overcome the complexity of the problem would be to find an approximation of
E [u(x, k)] using techniques, such as the Monte Carlo (MC) or spectral methods
(SM).
1.3 The Sampling Process
The sampling process consists in the selection of one (or more) individual, from
a well-defined set called a population. Each element is usually characterised by
a certain number of properties; these properties are allowed to discriminate the
population, according to specific criteria. For instance, we could be interested in
characterising a country according to the age of its citizens or according to their
educational level. We could be interested in identifying functioning/failed items
in a single batch from an industrial production line, or we could classify flowers
depending on the colour of their petals.
In descriptive statistics, the goal of the sampling process is to obtain a representative subset of individuals, the sample set, to estimate characteristics of
the whole population. Indeed, one single sample will give us all the information
about the random element we draw, but it will not tell us much about the whole
population. If a sufficient number (in the statistical sense) of samples are collected,
then the population can be characterised. The characterisation is strictly bound to
J. F. Reis et al.
Fig. 1.2 Plot of the function
in Eq. (1.6). This is the
solution of Eq. (1.5) when
k ∼ lognormal(0; 0, 1).
Space discritazation has 11
points
where u 0 and u 1 are the left and right hand-side boundary conditions. As stated
above, the solution plotted in Fig. 1.2 is obtained for the particular case k(x, θ) ≡
k ∼ lognormal(0, 0.1). In particular, the equality E[1/k] = 1/E[k] holds for such
distribution. The problem we are referring to is very simple, and there exist wellestablished procedures to compute the statistical moments of its solution. On the
other hand, if the problem was too complicated, it would be impossible to retrieve
the very same statistical information through an analytic procedure. The only way
to overcome the complexity of the problem would be to find an approximation of
E [u(x, k)] using techniques, such as the Monte Carlo (MC) or spectral methods
(SM).
1.3 The Sampling Process
The sampling process consists in the selection of one (or more) individual, from
a well-defined set called a population. Each element is usually characterised by
a certain number of properties; these properties are allowed to discriminate the
population, according to specific criteria. For instance, we could be interested in
characterising a country according to the age of its citizens or according to their
educational level. We could be interested in identifying functioning/failed items
in a single batch from an industrial production line, or we could classify flowers
depending on the colour of their petals.
In descriptive statistics, the goal of the sampling process is to obtain a representative subset of individuals, the sample set, to estimate characteristics of
the whole population. Indeed, one single sample will give us all the information
about the random element we draw, but it will not tell us much about the whole
population. If a sufficient number (in the statistical sense) of samples are collected,
then the population can be characterised. The characterisation is strictly bound to