Titel_SS06

Type II Extreme Maximum Value Distribution – Frechet max

For cumulative distribution functions downwards limited at zero and upwards unlimited with
a tail falling of in the form:

1

() 1

k

FxX  x

!

"#$% (2.76)

the cumulative distribution function of extreme maxima in the reference period T i.e.
FxXTmax, ()has the following form:

max

, ( ) exp( )

k

XT

u

Fx

x

!

"#$% (2.77)

with corresponding probability density function:

1

max

, ( ) exp( )

kk

XT

ku u

fx

ux x

!!

"#$% "#$% (2.78)

which is also denoted the Frechet distribution for extreme maxima. The mean value and the
variance of the Frechet distribution can be related to the parameters u and as: k

max

(^22) max 2

(1^1 )

2

(1 ) (1 )

T

T

X

X

u

k

u

kk





+ 

++

(^89) 

1

(2.79)

where it is noticed that the mean value only exists for and the standard deviation only
exist for. In general it can be shown that the i’th moment of the Frechet distribution
exist only when.

k& 1

k& 2

ki&

Type III Extreme Minimum Value Distribution – Weibull min

Finally in the case where the cumulative distribution function FxX() is downwards limited at
' and the lower tail falls of towards ' in the form:
Fx cx() (')k (2.80)

leads to a cumulative distribution function for the extreme minimum within the
reference period T of the following form:

min

FxXT, ()

min

, () 1 exp

k

XT

Fx x

u

'

'

! !

 "" "

$%$%

## (2.81)

with corresponding probability density function:

1

min

, () exp

kk

XT

fxkx x

uu u

'

'' '

 !

!!

"# "#""

$% $%$%

'

## (2.82)

which is also denoted the Weibull distribution for extreme minima. The mean value and the
variance of the Weibull distribution can be related to the parametersu, k and ' as: