Extreme Value distributions

Extreme value distributions are quite useful in civil engineering where the extreme conditions, which occur rarely, are the critical design elements. This is particularly true when considering the natural hazards such as the maximum intensity of the earth quake in the life span of a building, maximum flood levels in the life span of a bridge. These distributions are also useful in travel behavior modeling where the decisions related to the mode choice are presumed to follow extreme value distribution. Gumbel distribution, which is also known as the type 1 extreme value distribution, is one such distributions that deals with the extremely large values.

Figure 5.23: Gumbel distribution for the maximum extreme value with location parameter zero and the scale parameter 1

Gumbel distribution

This distribution results from the underlying assumption that the elements of a random sample follow exponential distribution. Cumulative distribution function for the exponential distribution is shown below;

Then from the marginal distribution for the maximum value of the random sample which is equals to

which results in,

With the introduction of the location and scale parameters b and a,

As n tends to infinity it becomes,

The corresponding probability density function would be;

for -∞<x<∞.

Gumbel distribution with location parameter 0 and the scale parameter is shown in Figure 5.23.