is defined over the range 
and approaches one as 
according to the relation.
Maximum Value Distribution
The type-II asymptotic distribution for the maximum
values is useful whenever the parent distribution is defined over the range and approaches one as according to the relation.
where, 
and 
are the parameters of the
distribution. The extreme value distribution for the maximum value, 
, is given by
The corresponding PDF is
where 
and
are the parameters of
the distribution; is
the characteristic maximum value of the underlying variable 
and , the shape parameter, is a measure of
dispersion. The Type-II asymptotic form is obtained as 
goes to infinity from an initial
distribution that has a polynomial tail in the direction of the extreme value,
which requires a polynomial tail. Therefore, a lognormal distribution converges
to a Type-II asymptotic form for the maximum value. However, the
Type-I converges from an exponential tail.
For the Type-II distribution of maxima, the mean,
standard deviation, and COV of 
are related to the distribution
parameter and as follows:
and
In these equations, 
is the gamma function. It is useful
for representing annual maximum winds and other meteorological and hydrological
phenomena.
Minimum Value Distribution
The Type-II asymptotic distribution for the minimum
values is useful whenever the parent distribution is defined over the range 
. In this case, the distribution
function for the minimum value, 
, is given by
where the parameter 
is the characteristic minimum value
of the initial variable and is the shape parameter, an inverse
measure of dispersion.
For the Type-II distribution of minima, the mean,
standard deviation, and COV of 
are related to the distribution
parameter and as follows:
and
In these equations, is the gamma function. It is not
commonly used since the required parent distribution shape is not commonly
observed in practical applications.