Fundamentals of Probability and Statistics for Engineers

The mean and variance associated with the Type-I maximum-value distribu-
tion can be obtained through integration using Equation (7.90). We have noted
that u is the mode of the distribution, that is, the value of y at which fY (y) is
maximum. The mean of Y is

where

It is seen from the above that u and are, respectively, the location and scale
parameters of the distribution. It is interesting to note that the skewness
coefficient, defined by Equation (4.11), in this case is

which is independent of and u. This result indicates that the Type-I
maximum-value distribution has a fixed shape with a dominant tail to the right.
A typical shape for fY (y) is shown in Figure 7.14.
The Type-I asymptotic distribution for minimum values is the limiting
distribution of Zn in Equation (7.91) as n from an initial distribution
FX(x) of which the left tail is unbounded and is of exponential type as it decreases
to zero on the left. An example of FX (x) that belongs to this class is the normal
distribution.
The distribution of Zn as n can be derived by means of procedures
given above for Yn through use of a symmetrical argument. Without giving
details, if we let

y

fY(y)

Figure 7.14 Typical plot of a Type-I maximum-value distribution

230 Fundamentals of Probability and Statistics for Engineers

0 577is Euler’s constant; and the variance is given by

mYˆu‡

;… 7 : 102 †

'

^2 Yˆ

^2

62

: … 7 : 103 †

1 ' 1 : 1396 ;

!1

!1

lim

n!1

ZnˆZ; … 7 : 104 †

: