with corresponding probability density function:
max
fXT, ( )xxux 55 exp( ( ) exp( 5 ( u))) (2.69)
which is also denoted the Gumbel distribution for extreme maxima. The mean value and the
standard deviation of the Gumbel distribution may be related to the parameters and u 5 as:
maxmax0.577216
6
TTXX uu^6
55
5
(2.70)
where 6 is Euler’s constant.
The Gumbel distribution has the useful property that the standard deviation is independent on
the considered reference period, i.e. XXmaxnT Tmaxand that the mean value XmaxnT depends on
in the following simple way:
nmax max max6
XXnT T XT ln( )n
(2.71)
Finally by manipulation of Equation (2.68) it can be shown, by utilising a Taylor expansion to
the first order of ln( )p inp 1 , that the characteristic value xc corresponding to an annual
exceedance probability of p and corresponding return period TpR1/ for a Gumbel max
distribution for large return periods can be written as:
(^1) ln( )
xcR 7 uT 5 (2.72)
which shows that the characteristic value, a typical engineering decision parameter, increases
with the logarithm of the considered return period.
Type I Extreme Minimum Value Distribution – Gumbel min
In case the cumulative distribution function is downwards unbounded and the lower
tail falls off in an exponential manner, symmetry considerations leads to a cumulative
distribution function for the extreme minimum within the reference period T of the
following form:
FxX()
min
FXT, ()x
min
FXT, ( ) 1 exp( exp( (x 5 xu))) (2.73)
with corresponding probability density function:
min
fXT, ( )xxux 55 exp( ( ) exp( ( 5 u))) (2.74)
which is also denoted the Gumbel distribution for extreme minima. The mean value and the
variance of the Gumbel distribution can be related to the parameters u and 5 as:
min
min
0.577216
6
TTXXuu