Fundamentals of Probability and Statistics for Engineers

the PDF of Z can be shown to have the form

where and u are again the two parameters of the distribution.
It is seen that Type-I asymptotic distributions for maximum and minimum
values are mirror images of each other. The mode of Z is u, and its mean,
variance, and skewness coefficients are, respectively,

F or probability calculations, values for probability distribution functions
FY (y) and FZ (z) over various ranges of y and z are available in, for example,
M icrosoft Excel 2000 (see Appendix B).

Ex ample 7. 9. Problem: the maximum daily gasoline demand Y during the
month of M ay at a given locality follows the Type-I asymptotic maximum-
value distribution, with mY 2and Y 1, measured in thousands of gallons.
Determine (a) the probability that the demand will exceed 4000 gallons in
any day during the month of May, and (b) the daily supply level that for 95%
of the time will not be exceeded by demand in any given day.
Answer: it follows from Equations (7.102) and (7.103) that parameters and
u are determined from

For part (a), the solution is

For part (b), we need to determine y such that

Some Important Continuous Distributions 231

FZ…z†ˆ 1 expfexp‰ …zu†Šg;1<z<1… 7 : 105 †

mZˆu

^2 Zˆ

^2

62

1 ' 1 : 1396

9

>>>

>=

>>

>>

;

… 7 : 106 †

ˆ  ˆ

ˆ





6

p

Y

ˆ





6

p ˆ 1 : 282 ;

uˆmY

0 : 577

ˆ 2 

0 : 577

1 : 282

ˆ 1 : 55 :

P…Y> 4 †ˆ 1 FY… 4 †

ˆ 1 expfexp‰ 1 : 282 … 4  1 : 55 †Šg

ˆ 1  0 : 958 ˆ 0 : 042 :

FY…y†ˆP…Yy†ˆ 0 : 95 ;