Sample final exam

CIVL 581 Final Examination
Problem 1
Circle T (true) or F (false) for each of the following statements. A correct answer will score 2
points; whereas one point will be deducted for each incorrect answer. The lowest possible total
score for problem 1 is zero.
If the return period of a traffic accident at Tsing Ma Bridge is 6 months, the
probability of a traffic accident occurring in Tsing Ma Bridge in a given year is 2.
Fault tree analysis may be used to obtain the probability of a branch event in an
event tree.
If two engineers disagree greatly on the probability value of a given event in a
decision tree, they may still end up selecting the same alternative, even if they both adopt
the same Maximum EMV criterion for decision.
Monte Carlo Simulation procedure can be considered for determining a
probability because it is convenient to perform, fast and accurate.
Preferential independence is an assumption to simplify Weighted Objective
Decision Analysis.
For a risk aversive person, it is more appropriate to adopt the Expected Utility
Value (EUV) criterion instead of the Expected Monetary Value (EMV) criterion for
choosing optimal alternative.
Probabilistic Observational Method can be used to verify if a given set of
observed data is better modeled by a lognormal distribution instead of a normal
For the Bernoulli Sequence Model, the time between two successive occurrences
will definitely not follow an exponential distribution.
The failure probability of a system can be smaller than the failure probability of
any of its components.
Problem 2
If Events A and B are statistically independent, P(AUB)= P(A)+P(B).
Two options are available to build a railroad between two towns, either by constructing an
underground tunnel (U), or by building on ground surface (G). If option G is used, the project
completion time T (in months) would be randomly distributed with mean = 10 and standard
deviation = 1.5. The efficiency of tunnel construction, however, depends on the soil condition
(assumed uniform throughout the volume to be drilled): if the soil is soft, then T has a mean of 5
and standard deviation = 2, but for hard soil the mean completion time would be 20 with a
standard deviation of 5. There is 20% chance of encountering hard soil for the construction
region. For parts (a), (b) and (c) below, assume that the railroad company is only concerned with
the project completion time, and money is not a problem.
(a) Which option should be used if the decision criterion is to minimize expected completion
(b) Suppose the company’s utility function for time is u(t) = -5t2. Which option should be chosen
to maximize expected utility?
(c) How much additional time (note that this would increase the completion time), at most,
should the company be willing to spend to gather information on the soil condition? Assume
the same criterion as in (b).
(d) Other than time, what additional factors may be considered in this decision analysis problem?
Problem 3 (15%)
A newly developed construction material was tested for its fire resistance. Results from testing
two samples show that one had damages after exposed to only 2 hours of fire, while the other
could last for 3 hours. Suppose the duration of fire, T, for damage of a material follows a normal
distribution N(, 0.5). Before the tests, the mean duration until damage, for this material was
expected to be 3 hours with a c.o.v. of 20% based on a theoretical study.
Determine the updated mean and c.o.v. of the mean duration until damage for this
With the above available information, what is the probability that a wall constructed
using this material will be damaged in less than 1 hour of fire?
Problem 4
Accidents during the operation of a nuclear power plant may involve the leakage of radioactive
materials, which could be hazardous to the immediate neighborhood. Let X denote the radial
distance (in km) from the power plant beyond which health hazards will not exist after an
accident. The probability density of X is uniformly distributed as
fX(x) = 1/h
where h is the parameter denoting the maximum possible dispersion distance of the radioactive
material. Based on theoretical studies, a prior distribution of h has been established as
f’H(h) = 0.003 h2
0 < h < 10
(a) Determine the expected distance of the hazardous zone based on the given information.
(b) Suppose an accident has occurred and the hazardous zone was observed to be within a radius
of 4 km.
What is the range of H now?
Determine the updated distribution of H.
(iii) What is the probability that the hazardous zone will not exceed 4 km in radius in the
next accident?
Problem 5 (25%)
Y, M and T are three uncorrelated random variables defining the performance function of an
engineering system, with respective means 6500, 20 and 15 (in suitable units). All variables are
normally distributed, with c.o.v.’s 0.15, 0.2 and 0.1, respectively. The performance would be
unsatisfactory (i.e. fail) if the performance function
G = Y2/4 – (250/)2 (M2 + T2)
becomes negative.
(a) Calculate the mean and standard deviation of G using the first order approximation.
(b) Compute the probability of unsatisfactory performance of the system based on results from
(a) and the assumption that G is normally distributed.
(c) Determine the reliability index  for this system using an appropriate iteration technique.
Perform two iterations and present your answers in table form; report your final  value to
three decimal places.
(d) Compute the failure probability of the system based on the reliability index found in (c).
(e) Do your answers in (b) and (d) agree with each other? Which do you think is a better answer,
and why?
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