Fundamentals of Probability and Statistics for Engineers

where u is the scale factor and the value of u describes the characteristics of
a river; it varies from 1.5 for violent rivers to 10 for stable or mild rivers.
In closing, let us remark again that the Type-I maximum-value distribution
is valid for initial distributions of such practical importance as normal, lognor-
mal, and gamma distributions. It thus has wide applicability and is sometimes
simply called the extreme value distribution.

7.6.2 Type-II Asymptotic Distributions of Extreme Values

The Type-II asymptotic distribution of maximum values arises as the limiting
distribution of Yn as n from an initial distribution of the Pareto type, that
is, the PDF FX(x) of each Xj is limited on the left at zero and its right tail is
unbounded and approaches one according to

For this class, the asymptotic distribution of Yn,FY (y), as n takes the
form

Let us note that, with FX(x) given by Equation (7.111), each Xj has moments
only up to order r, where r is the largest integer less than k. If k > 1, the mean of
Yis

and, if k > 2, the variance has the form

The derivation of FY (y) given by Equation (7.112) fo llows in broad outline
that given for the Type-I maximum-value asymptotic distribution and will not
be presented here. It has been used as a model in meteorology and hydrology
(G umbel, 1958).
A close relationship exists between the Type-I and Type-II asymptotic
maximum-value distributions. Let YI and YII denote, respectively, these random

Some Important Continuous Distributions 233

!1

FX…x†ˆ 1 axk; a;k> 0 ;x 0 : … 7 : 111 †

!1

FY…y†ˆexp

y

v

k

; v;k> 0 ;y 0 : … 7 : 112 †

mYˆv 1 

1

k



; … 7 : 113 †

^2 Yˆv^2  1 

2

k



^21 

1

k



: … 7 : 114 †