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

where parameters and u in FYI (y) arerelated to parametersk and vin FYII (y) by

When they are continuous, their pdfs obey the relationship

The Type-II asymptotic distribution of minimum values arises under analogous
conditions. With PDF FX(x) limited on the right at zero and approaching zero
on the left in a manner analogous to Equation (7.111), we have

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 Zn as n for an initial distribution FX (x) in, which the left tail
increases from zero near x in the manner

This class of distributions is bounded on the left at The gamma distri-
bution is such a distribution with 0.
Again bypassing derivations, we can show the asymptotic distribution for the
minimum value to be

234 Fundamentals of Probability and Statistics for Engineers

FYII…y†ˆFYI…lny†;y 0 ;… 7 : 115 †

uˆlnvand ˆk:… 7 : 116 †

fYII…y†ˆ

1

y

fYI…lny†; y 0 : … 7 : 117 †

FZ…z†ˆ 1 exp

z

v

k

; v;k> 0 ;z 0 : … 7 : 118 †

!1

FX…x†ˆc…x"†k; c;k> 0 ;x": … 7 : 119 †

xˆ".

"ˆ

FZ…z†ˆ 1 exp

z"

w"

k

; k> 0 ;w>";z"; … 7 : 120 †

ˆ"