Page 251 - Fundamentals of Probability and Statistics for Engineers

234 Fundamentals of Probability and Statistics for Engineers

variables. It can be verified, using the techniques of transformations of random
variables, that they are related by

ln y†; y 0; 7:115†
F Y II y† ˆ F Y I
where parameters and u in F Y I (y) are related to parameters k and v in F Y II (y) by

u ˆ ln v and ˆ k: 7:116†

When they are continuous, their pdfs obey the relationship
1
f y†ˆ f ln y†; y 0: 7:117†
y
Y II Y I
The Type-II asymptotic distribution of minimum values arises under analogous
conditions. With PDF F X (x) limited on the right at zero and approaching zero
on the left in a manner analogous to Equation (7.111), we have

k
; v; k > 0; z 0: 7:118†
z
v
F Z z†ˆ 1 exp
However, it has not been found as useful as its counterparts in Type I and Type III
as in practice the required initial distributions are not frequently encountered.

7.6.3 TYPE-III ASYMPTOTIC DISTRIBUTIONS OF EXTREME
VALUES

Since the Type-III maximum-value asymptotic distribution is of limited prac-
tical interest, only the minimum-value distribution will be discussed here.
The Type-III minimum-value asymptotic distribution is the limiting distribu-
tion of Z n as n !1 for an initial distribution F X (x) in, which the left tail
increases from zero near x ˆ " in the manner

k
F X x†ˆ c x "† ; c; k > 0; x ": 7:119†
This class of distributions is bounded on the left at x ˆ ". The gamma distri-
bution is such a distribution with " ˆ 0.
Again bypassing derivations, we can show the asymptotic distribution for the
minimum value to be

z "

k
F Z z†ˆ 1 exp ; k > 0; w >"; z "; 7:120†
w "

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