Page 243 - Fundamentals of Probability and Statistics for Engineers

226 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. Following Equation (5.12), we have

8
1 ‡ †
> 1 1
< y a† b y† ; for a x b;
f y†ˆ b a† ‡ 1 † † 7:86†
Y
>
0; elsewhere:
:
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 X j , j ˆ 1, 2, .. . , n, denote the jth gust velocity of n gusts
occurring in a year, and let Y n denote the annual maximum gust velocity. We
are interested in the probability distribution of Y n in terms of those of X j . In the
following development, attention is given to the case where random variables
X j , j ˆ 1, 2, .. . , n, are independent and identically distributed with PDF F X (x)
and pdf f (x) or pmf p (x). Furthermore, asymptotic results for n !1 are
X
X
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 Z n
of random variables X 1 , X 2 ,..., and X n , which is also of interest in practical
applications.
The random variables Y n and Z n are defined by

Y n ˆ max X 1 ; X 2 ; ... ; X n †;
7:87†
Z n ˆ min X 1 ; X 2 ; ... ; X n †:

The PDF of Y n is

y†ˆ P Y n y†ˆ P all X j y†
F Y n
ˆ P X 1 y \ X 2 y \ \ X n y†:

TLFeBOOK