Titel_SS06

In the same way the cumulative distribution function of the largest extreme in a period of ,
may be determined from the cumulative distribution function of the largest extreme
in the period T, , by:

nT

max

FxXnT, ()

max

max

FxXT, ()

max n

FxFxXnT,,() XT() (2.65)

which follows from the multiplication law for independent events. The corresponding
probability density function may be established by differentiation of Equation (2.65) yielding:
max max 1 max
,,() () ()

n

fXnTxnF x f xXT XT,

  (2.66)

In Figure 2.16 the case of a Normal distribution with mean value equal to 10 and standard
deviation equal to 3 is illustrated for increasing n.

0

0.10

0.20

0.30

5101520

x

0

0.40

n=1

n=5

n=10

n=50

n=100

n=500

f (x)

Figure 2.16: Normal extreme value probability density functions.

Similarly to the derivation of Equation (2.65) the cumulative distribution function for the
extreme minimum value in a considered reference period T, FxXnTmin, () may be found as:
min min
, () 1 (1 ())

n

FxXnT  FxXT, (2.67)

Subject to the assumption that the considered process is ergodic it can be shown that the
cumulative function for an extreme event FXnTmax, ()x converges asymptotically (as the reference
period increases) to one of three types of extreme value distributions, i.e. type I, type II or
type III. To which type the distribution converges depends only on the tail behaviour (upper
or lower) of the considered random variable generating the extremes, i.e.

nT

max
FXT, ()x. In the
following the three types or extreme value distributions will be introduced and it will be
discussed under what conditions they may be assumed. In Table 2.8 the definition of the
extreme value probability distributions and their parameters and moments is summarised.

Type I Extreme Maximum Value Distribution – Gumbel max

For upwards unbounded distribution functions where the upper tail falls off in an
exponential manner such as it is the case for the exponential function, the Normal distribution
and the Gamma distribution the cumulative distribution of extremes in the reference period T
i.e. has the following form:

FxX()

max

FxXT, ()

max

FxXT, ( ) exp( exp( 5 (xu))) (2.68)