Figure 2.15: Time series of recorded daily maximal water levels together with 5 days (arbitrary point in
time) and 10 days maximal sample frequency histograms of water levels.
In the following some results are given concerning the extreme events of trials of random
variables and random processes, see also Madsen et al. (1986) and Benjamin and Cornell
(1969). Taking basis in the tail behaviour of cumulative distribution functions asymptotic
results are given leading to the so-called extreme value distributions.
Extreme Value Distributions
When extreme events are of interest the arbitrary point in time distribution of the load variable
is not of immediate relevance but rather the distribution of the maximal values of the
considered quantity over a given reference period.
If the random process X()t may be assumed to be ergodic (see the lecture notes Basic Theory
of Probability and Statistics in Civil Engineering, Faber, 2006) the distribution of the largest
extreme in a reference period T, can be thought of as being generated by sampling
values of the maximal realisation
max
FxXT, ()
xmax from successive reference periods T. If the values of
xmax are represented by the random variable Y, the cumulative distribution function is
the cumulative distribution function of the extreme maximum realisation corresponding to the
considered reference period.
FyY()T