Titel_SS06

(^) -1(F (x))X
x
F (x)X
Figure 2.20: Concrete cube compressive strength data plotted in Normal distribution paper.
When using probability paper for the consideration of extreme phenomena such as e.g. the
maximum water level in a one year period the probability paper may also be used for the
purpose of estimating the values of the water level with a certain return period i.e. for the
purpose of extrapolation (see e.g. Schneider, 1994). However, as always when extrapolating,
extreme care must be exercised.
2.13 Estimation of Distribution Parameters
There are in principle two different methods to estimate the distribution parameters on the
basis of data, namely the methods of point estimates and the methods of interval estimates. In
the following, however, only two of the methods of point estimates will be explained, namely
the method of moments and the method of maximum likelihood as these have proven especially
useful in practical risk and reliability engineering analysis.
The Method of Moments
Assuming that the considered random variable X may be modelled by the probability density
function ( ; )fX x where ( , ,..,:: : 12 k)Tare the distribution parameters, the first moments k
12 ,.., )
T
(,(( (k of the random variable X may be written as:
12

() ( )

( , ,.., )

j

jX

jk

( x fx dx

(:: :

   





 (2.88)

If the random sample that will be used to estimate the distribution parameters
( , ,.., 12 )
T
:: :k are collected in the vector ˆˆˆ ˆ( ,12,,.., )
T
x x xxn the corresponding sample
moments may be calculated as:

k

1

(^1) ˆ

 
n j
j
i
m
n i
x (2.89)