Page 248 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 231

the PDF of Z can be shown to have the form

F Z z†ˆ 1 expf exp‰ z u†Šg; 1 < z < 1 7:105†

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,

9
m Z ˆ u >
>
>
>
=
2 7:106†
2
ˆ
Z 2 >
6 >
>
>

1 ' 1:1396 ;
For probability calculations, values for probability distribution functions
F Y (y) and F Z (z) over various ranges of y and z are available in, for example,
Microsoft Excel 2000 (see Appendix B).

Example 7.9. Problem: the maximum daily gasoline demand Y during the

month of May at a given locality follows the Type-I asymptotic maximum-
value distribution, with m Y ˆ 2 and Y ˆ 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

ˆ p  ˆ p ˆ 1:282;

6 Y 6
0:577 0:577
u ˆ m Y ˆ 2 ˆ 1:55:
1:282
For part (a), the solution is
P Y > 4†ˆ 1 F Y 4†
ˆ 1 expf exp‰ 1:282 4 1:55†Šg
ˆ 1 0:958 ˆ 0:042:

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

F Y y†ˆ P Y y†ˆ 0:95;

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