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CEE 304 – UNCERTAINTY ANALYSIS IN ENGINEERING
2005 Final Examination
2:00 – 4:30 PM, Friday, December 16, 2005
Exam is open book and open notes.
It lasts 150 minutes and there are 150 points.
SHOW WORK!
1.
(10 pts – 2 pts each) Short answer and true/false questions.
WRITE ANSWERS IN EXAM BOOK.
(a)
True or False
: The joint density function f
XY
(x,y) can always be obtained as the
product f
X
(x)f
Y
(y).
(b)
Given two independent events A and B, the general formula needed to compute
P(A
∪
B) in terms of P(A) and P(B) is _______________________.
(c)
The sample average computed with n observations has a standard error of ___________.
(d)
True or False
: By the Central Limit Theorem, the estimator
x
of the mean is always
normally distributed, assuming the sample is large enough.
(e)
True or False
: Maximum likelihood estimators are always unbiased.
2.
(6 pts) A charcoal filter can remove chlorinated hydrocarbons from drinking water. An
environmental engineer collected 12 independent samples of water that passed through such a
filter. The average of the 12 observed removal rates was
x
= 99.27% with sample standard
deviation s = 0.25%. Assuming that the measurements are normally distributed about the true
removal rate
μ
,
(a)
What is 99% confidence interval for
μ
?
(b)
Does the interval computed above indicate that precise information about the mean
μ
is
available?
3.
(12 pts) This time of year there are Christmas lights everywhere. The bulbs seem to burn out at
random times. Suppose that the system of lights decorating the home could be described by a
Poisson Process, wherein the arrival rate of failures is 0.2 per hour.
(a)
What is the probability that no lights fail in 3 hours of operation?
(b)
During a week (50 hours of operation), what is the mean and variance of the number of
failures that will occur?
(c)
What is the mean and variance of the time (operational) until the system experiences 4 failures?
(d)
What is the probability 10 hours (operational hours) pass before 2 lights fail?
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View Full DocumentCEE 304 – UNCERTAINTY ANALYSIS IN ENGINEERING
2005 Final Examination
2:00 – 4:30 PM, Friday, December 16, 2005
4.
(16 pts) A structural engineer is concerned with the possible wind force on the side of a tall
building. The total force F depends on the wind speed W, and the angle of the wind which
determines the effective area A, such that F = W*A. The mean and standard deviation of W are
160 and 120. [Thus, the mean and standard deviation of ln(W) are 4.85 and 0.668.] The effective
area A has median 50 and coefficient of variation 0.3, Assuming that W and A are independent
random variables, and are both lognormally distributed, what are the median, mean, and standard
deviation of the total force F on the structure?
5.
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 Fall '08
 Stedinger

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