Fundamentals of Probability and Statistics for Engineers

or

Taking logarithms of Equation (7.107) twice, we obtain

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 yT denote the maximum flood associated with return
period T. Determine the relationship between yT 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 gr eater than yT.
Answer: assuming that floods occur independently, the number of years
between floods with magnitudes greater than yT assumes a geometric distribu-
tion. Thus

N ow, from Equation (7.101),

where b (yT u). The substitution of Equation (7.109) into Equation
(7.108) gives the required relationship.
For values of yT where FY (yT ) 1, an approximation can be made by
noting from Equation (7.109) that

Since FY (yT ) is close to 1, we retain only the first term in the foregoing
expansion and obtain

Equation (7.108) thus gives the approximate relationship

232 Fundamentals of Probability and Statistics for Engineers

expfexp‰ 1 : 282 …y 1 : 55 †Šgˆ 0 : 95 : … 7 : 107 †

yˆ 3 : 867 ;

Tˆ

1

P…Y>yT†

ˆ

1

1 FY…yT†

: … 7 : 108 †

FY…yT†ˆexp‰exp…b†Š; … 7 : 109 †

ˆ 

!

exp…b†ˆlnFY…yT†ˆf‰FY…yT† 1 Š

1

2

‰FY…yT† 1 Š^2 ‡g:

1 FY…yT†'exp…b†:

yTˆu 1 ‡

1

u

lnT



; … 7 : 110 †