Page 249 - Fundamentals of Probability and Statistics for Engineers

232 Fundamentals of Probability and Statistics for Engineers

or
expf exp‰ 1:282 y 1:55†Šg ˆ 0:95: 7:107†

Taking logarithms of Equation (7.107) twice, we obtain

y ˆ 3:867;

that is, the required supply level is 3867 gallons.
Example 7.10. Problem: consider the problem of estimating floods in the
design of dams. Let y T denote the maximum flood associated with return
period T. Determine the relationship between y T and T if the maximum river
flow follows the Type-I maximum-value distribution. Recall from Example 6.7
(page 169) that the return period T is defined as the average number of years
between floods for which the magnitude is greater than y T .
Answer: assuming that floods occur independently, the number of years
between floods with magnitudes greater than y T assumes a geometric distribu-
tion. Thus

1 1
T ˆ ˆ : 7:108†
P Y > y T † 1 F Y y T †

Now, from Equation (7.101),
F Y y T †ˆ exp‰ exp b†Š; 7:109†

where b ˆ (y T u). The substitution of Equation (7.109) into Equation
(7.108) gives the required relationship.
For values of y T where F Y (y T ) ! 1, an approximation can be made by
noting from Equation (7.109) that

1 2
exp b†ˆ ln F Y y T † ˆ f‰F Y y T † 1Š ‰F Y y T † 1Š ‡ g:
2
Since F Y (y T ) is close to 1, we retain only the first term in the foregoing
expansion and obtain

1 F Y y T †' exp b†:

Equation (7.108) thus gives the approximate relationship

1
y T ˆ u 1 ‡ ln T ; 7:110†
u

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