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under consideration; thus to make the probability paper generally applicable to any data
set it is necessary to eliminate the dependency on these parameters. This can be done
with a linear transformation of the type
y
=
a ( x
b )
(13.27)
in which the nature of a and b depends on the particular function F ( x ) that is being
used. For instance, in the case of the normal distribution one has b
= σ 1 .In
the case of the first asymptote for largest values, as will be seen below, b i s the mode,
that is the value of x where the density f ( x ) is a maximum, and a
= μ
and a
π/ 6)
σ 1 . This
=
(
transformation results in a distribution function, say F Y ( y ), given by
F Y ( y )
=
F ( b
+
y
/
a )
(13.28)
which is independent of the parameters. In principle, probability paper is constructed as a
plot of x versus y , with the values of a and b left unspecified; when F ( x ) is symmetrical,
y
0 is placed at the center of the scale. Parallel to the y -scale, a F Y ( y )-scale is plotted
and in some types of probability paper a third scale is shown with T r ( y )[
=
F Y ) 1 ]
values. However, most types of probability paper do not show the y -scale, and only
one scale, either F Y ( y )or T r ( y ), is displayed. While normal probability paper, which
is based on the normal probability distribution, has been made available commercially
in the past, nowadays normal scales can be generated by standard computer programs.
For some applications, the x -variable is transformed logarithmically, so that it is log( x )
which is plotted against F Y ( y )or T r ( y ). Figures 10.34-10.37 illustrate applications of
lognormal probability paper.
=
(1
Example 13.1. Probability paper based on an extreme value distribution
The first asymptotic distribution for largest values, which is treated in detail below in
Section 13.4.5, can be used here to illustrate the construction of probability paper. This
function is given by F ( x )
y )] in which y is the linearly scaled or reduced
variable defined in Equation (13.27). This distribution can be immediately inverted to
yield y
=
exp[
exp(
ln[ ln( F 1 )] and y
1)]]. One starts with graph paper in
which one of the coordinate axes is designated as the y coordinate axis. Values of y can
then be calculated for selected values of F covering the entire F range of interest, and
marked with their F value on the y -axis, or (which is preferable to avoid clutter) on a
separate axis parallel to y . The same is done for selected T r values covering the entire
T r range of interest. As mentioned, on most types of probability paper not all three axes
are shown, but only the resulting F -or T r -axis. Figure 13.3 shows a lay-out with a y -axis
and a T r -axis.
=−
=−
ln [ ln[ T r /
( T r
13.2.5
Theoretical probability distribution functions
Countless mathematical functions are consistent with the definition of probability given
above, which can be used to describe data sets. In frequency analysis, it is often useful
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