Problem solving paper- EngineeringQuestion – 1
The Arena costs of $900million to build. The owner wants to insure it. The probability
distribution of total annual damages ݂ሺݔሻ ൌ
ఒ ቀ
௫
ఒ
ቁ
ିଵ
eሺ୶⁄ ሻ ఒ ೖ
in which,
x = annual loss in 108 dollars
the shape parameter, ݇ ൌ 1.5
the scale parameter, ߣ ൌ 6
The statistical properties of the Weibull PDF are given in Error! Reference source not found..
The mean of this PDF is, thus, ~5.4 and the standard deviation is ~3.7.
Figure Error! No text of specified style in document..1. Probability distribution of annual loss and utility
for loss.
Utility of loss. The corporate utility function of annual loss has been approximated as
ݑሺݔሻ ൌ 1 െ exp ሺ0.3xሻ
What is the maximum insurance premium the owner should pay to fully cover any risk.
Question – 2
Badger Enterprises publishes a satirical magazine, Badger Zone™, which specializes in
publishing embarrassing details of the private lives of politicians (Figure 3.1). The
editors are aware of a scandalous rumor about a member of Congress and wish to
proceed with the story in a manner that maximizes expected net income. The risk of an
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
‐450.00
‐400.00
‐350.00
‐300.00
‐250.00
‐200.00
‐150.00
‐100.00
‐50.00
0.00
0.00 5.00 10.00 15.00 20.00
PDF, f(x)
Utility, u(x)
Annual loss, x (100m)
u(x) f(x)
expensive libel action cannot be ignored, and the editors estimate that the effects of
such an action will be a cost of $125,000 net if lost, taking into account possible fines,
legal costs, effects on future sales, and so on, and zero if won. These costs, and the
probability of losing, estimated to be 0.2, are, they believe, independent of whether or
not the rumor is true. However, the probability of the politician suing does depend on
whether the rumor is true, and they estimate that the probability of his taking legal action
is 0.3 if the rumor is true and 0.9 if it is not. They currently estimate the probability of the
rumor being true at 0.5.
They have three possible courses of action:
(1) Dropping the story,
(2) Publishing immediately, or
(3) Hiring a private detective to investigate the rumor further so that after his
report they
could take a decision on whether or not to publish.
The private detective they have in mind would cost $10,000, and he is good but not
infallible.
The probability of his saying the rumor is true is 1.0 if indeed it is true and 0.2 if it is
false. He will definitely answer in one direction or the other, but the probability of being
sued depends only on whether the rumor is true or not.
They estimate extra net income (gross of any costs of legal activity) from increased
circulation if they publish to be $50,000, and this would be independent of the truth of
the story. What should they do? How much should they be prepared to pay for a report
from an infallible private detective
