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 Yn (as ) from an initial distribution F (^) X(x) of
which the right tail is unbounded and is of an exponential type; that is, FX(x)
approaches 1 at least as fast as an exponential distribution. For this case, we
can express FX(x) in the form
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
We have the following important result (Theorem 7.6).
Theorem 7. 6: let random variables X 1 ,X 2 ,..., and Xn be independent and
identically distributed with the same PDF FX (x). If FX (x) is of the form given
by Equation (7.93), we have
where 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 fir st define a quantity un, known as the characteristic value of Yn,by
It is thus the value of Xj,j 1,2,...,n, at which P(Xj un) 1 1/n. As n
becomes large, FX(un) approaches unity, or, un is in the extreme right-hand tail
of the distribution. It can also be shown that un is the mode of Yn, which can
be verified, in the case of Xj being continuous, by taking the derivative of fYn(y)
in Equation (7.90) with respect to y and setting it to zero.
228 Fundamentals of Probability and Statistics for Engineers
n!1
FX…x†ˆ 1 exp‰g…x†Š; … 7 : 93 †
lim
n!1
YnˆY: … 7 : 94 †
FY…y†ˆexpfexp‰ …yu†Š; 1<y< 1 ; … 7 : 95 †

">0)

FX…un†ˆ 1 

1

n

: … 7 : 96 †

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