Page 250 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 233

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 Y n as n !1 from an initial distribution of the Pareto type, that
is, the PDF F X (x) of each X j is limited on the left at zero and its right tail is
unbounded and approaches one according to

k
F X x†ˆ 1 ax ; a; k > 0; x 0: 7:111†
For this class, the asymptotic distribution of Y n , F Y (y), as n !1 takes the
form

y k

F Y y†ˆ exp ; v; k > 0; y 0: 7:112†
v
Let us note that, with F X (x) given by Equation (7.111), each X j has moments
only up to order r, where r is the largest integer less than k. If k > 1, the mean of
Y is

1

m Y ˆ v 1 ; 7:113†
k
and, if k > 2, the variance has the form

2 2 1
2
2
ˆ v 1 1 : 7:114†
Y
k k
The derivation of F Y (y) given by Equation (7.112) follows 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
(Gumbel, 1958).
A close relationship exists between the Type-I and Type-II asymptotic
maximum-value distributions. Let Y I and Y II denote, respectively, these random

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