is not bounded, which is of
an exponential type. In such a case, the distribution function can be expressed
as
Maximum Value Distribution
This distribution is, referred to as the Gumbel
distribution, useful when the right tail of the parent distribution is not bounded, which is of
an exponential type. In such a case, the distribution function can be expressed
as
where 
increases with x monotonically. The
distributions such as normal, log-normal and gamma distributions belong to this
category. The extreme value distribution for the maximum value, 
, is given by
where the parameters of distribution, 
and 
, can be determined from the observation
data. They are related to the mean and the standard deviation of the extreme
value 
as
and
Where 
is the Euler’s constant.
Minimum Value Distribution
This distribution is useful whenever the left tail of the
parent distribution is unbounded and decreases to zero towards the left in an
exponential form. In this case, the distribution function for the minimum value,
, is given by
where, 
and 
are the parameters of the
distribution. They are related to the mean and the standard deviation of the
extreme value 
as
and
Where is the Euler’s constant.