Page 245 - Fundamentals of Probability and Statistics for Engineers

228 Fundamentals of Probability and Statistics for Engineers

to as Gumbel’s extreme value distribution, and included in Type III is the
important Weibull distribution.

7.6.1 TYPE-I ASYMPTOTIC DISTRIBUTIONS OF EXTREME
VALUES

Consider first the Type-I asymptotic distribution of maximum values. It is the
limiting distribution of Y n (as n !1 ) from an initial distribution F X (x) of
which the right tail is unbounded and is of an exponential type; that is, F X (x)
approaches 1 at least as fast as an exponential distribution. For this case, we
can express F X (x) in the form

F X x†ˆ 1 exp‰ g x†Š; 7:93†

where g(x) is an increasing function of x. A number of important distributions
fall into this category, such as the normal, lognormal, and gamma distributions.
Let

lim Y n ˆ Y: 7:94†
n!1
We have the following important result (Theorem 7.6).

Theorem 7.6: let random variables X 1 , X 2 , .. ., and X n be independent and

identically distributed with the same PDF F X (x). If F X (x) is of the form given
by Equation (7.93), we have
F Y y†ˆ expf exp‰ y u†Š; 1 < y < 1; 7:95†

where " > 0) and u are two parameters of the distribution.
Proof of Theorem 7.6: we shall only sketch the proof here; see Gumbel (1958)
for a more comprehensive and rigorous treatment.
Let us first define a quantity u n , known as the characteristic value of Y n , by

1
F X u n †ˆ 1 : 7:96†
n
ˆ
It is thus the value of X j , j 1, 2, ..., n, at which P(X j u n ) ˆ 1 1/n. As n

becomes large, F X (u n ) approaches unity, or, u n is in the extreme right-hand tail
of the distribution. It can also be shown that u n is the mode of Y n , which can
be verified, in the case of X j being continuous, by taking the derivative of f (y)
Y n
in Equation (7.90) with respect to y and setting it to zero.

TLFeBOOK