Fundamentals of Probability and Statistics for Engineers

developed in Chapter 5 can be applied to determine the pdf of Y in a straight-
forward manner. F ollowing Equation (5.12), we have

7.6 Extreme-Value Distributions

A structural engineer, concerned with the safety of a structure, is often inter-
ested in the maximum load and maximum stress in structural members. In
reliability studies, the distribution of the life of a system having n components
in series (where the system fails if any component fails) is a function of the
minimum time to failure of these components, whereas for a system with a
parallel arrangement (where the system fails when all components fail) it is
determined by the distribution of maximum time to failure. These examples
point to our frequent concern with distributions of maximum or minimum
values of a number of random variables.
To fix ideas, let Xj,j 1,2,...,n, denote the jth gust velocity of n gusts
occurring in a year, and let Yn denote the annual maximum gust velocity. We
are interested in the probability distribution of Yn in terms of those of Xj.Inthe
following development, attention is given to the case where random variables
Xj,j 1,2,...,n, are independent and identically distributed with PDF FX(x)
and pdf fX(x) or pmf pX (x). Furthermore, asymptotic results for are
our primary concern. For the wind-gust example given above, these conditions
are not unreasonable in determining the distribution of annual maximum gust
velocity. We will also determine, under the same conditions, the minimum Zn
of random variables X 1 ,X 2 ,..., and Xn, which is also of interest in practical
applications.
The random variables Yn and Zn are defined by

The PDF of Yn is

226 Fundamentals of Probability and Statistics for Engineers

fY…y†ˆ

1

…ba†^ ‡^ ^1

… ‡ †

… †… †

…ya†^ ^1 …by†^ ^1 ; foraxb;

0 ; elsewhere:

8

><

>:

… 7 : 86 †

ˆ

ˆ

n!1

Ynˆmax…X 1 ;X 2 ;...;Xn†;

Znˆmin…X 1 ;X 2 ;...;Xn†:

… 7 : 87 †

FYn…y†ˆP…Yny†ˆP…allXjy†

ˆP…X 1 y\X 2 y\\Xny†: