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CEE 304

UNCERTAINTY ANALYSIS IN ENGlhlEERlNG
Final Exam
Saturday, December 15,1989
This exam is open notes and openbook.
The exam lasts 150 minutes and there are 150 points.
SHOW WORK!
1. (15 points) An engineer believes that a random variable X has a cumulative
distribution function
FX(x)
=
1

exp(

kx3
}
for x
2
0
Given a random sample of n independent observations
(
Xi,
X2,
...
,
Xn},
what is
the maximum likelihood estimator of k? Why do statisticians like maximum
likelihood estimators?
2. (10 points)
In a region, annual maximum rainfall depths are modeled by a
lognormal distribution with a coefficient of variation of 0.20 and a median value
of 3 inches. What is the depth that would be exceeded with a probability of only
2%?
he
CV is
o,/p,
where X is the realspace random variable.)
3.
(10 points) Consider the seven observations
127
106
115
60
87
78
114
which are thought to be drawn from a normal distribution. By hand, or using
the attached probability paper, construct a probability plot of the data as a visual
test of normality. Provide a table of the two coordinates of the points you
plotted.
How
should one construct a probability plot to examine whether a sample
was drawn from an exponential distribution? What would one plot against
what?
4.
(15 points) Global climate change is a big issue now.
A
meterologist is trying
to track the concentration of C02 in the atmosphere.
Last week,
14
measurements of atmospheric C02 were made at his station; the observations
have a sample average of 334.12 ppm with a sample variance of 0.14 (ppm)2.
What is a 95% confidence interval for the true atmospheric concentration of C02
that week? (Assume that the variations are normally distributed about the true weekly mean
concentration.)
If the meterologist repeats his analysis every week, what is the probability
that
every
one of the confidence interval he constructs over the next 52 week
contains the true atmospheric C02 concentration for its week? What is the
probability the particular confidence interval you just constructed with the given
data contains the true atmospheric C02 concentration for last week?
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304

UNCERTAINTY ANALYSIS IN ENGINEERING
Final Exam
Saturday, December 15,1989
5.
(20 points) Two fraternities got into an argument about which is better
athletically.
They decided to resolve the issue by randomly pairing off members
of the two houses (denoted
r
and A) for 50yard races.
They conducted 25 races
and saw who won.
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 Fall '08
 Stedinger

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