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1 Bulbs (35 points)
A building contains 1000 light bulbs. The life (in days) of a bulb is random
variable with normal distribution N (µ , sigma). The
supplier claims that µ = 100 days and sigma
= 20 days. The cost of replacing a single bulb is:
50 cents if it is burnt out,
10 cents if the bulb is replaced before burning out,
in addition to the cost of the bulb, 80 cents.
Management wants to test a policy of replacing all bulbs after T days,
while continuing to replace bulbs that have expired before the scheduled
replacement. You were hired as consultant.
(a) Use simulation to determine an optimal policy, and 95% confidence
interval for the total cost.
(b) Using the policy found in (a), do a 2-way sensitivity analysis
of the average cost for values of µ and sigma
around the nominal values.
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2 Investment (35 points)
You currently have $10000. Each month you can invest any amount of money
you currently have in risky bonds. With probability 0.3 the amount you
invest is doubled (for example, if you invest $100, you get it back, plus
another $100). Also, with probability 0.7, the amount you invest is lost.
Consider the following investment strategies:
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Each month invest 15% of your money.
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Each month invest 35% of your money.
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Each month invest 50% of your money.
(a) Use simulation to determine which investment policy would be
the best for a 100 months period.
(b) Consider the following rule for stopping (even before 100
months):
Stop if your capital falls below $2000 or goes above $20000.
Estimate, for each of the above strategies, the number of months until
the game stops, and the final capital.
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3 Gambling (30 points) Jane
Smith plans on going to Atlantic City to play the quarter slot machines.
She will start with $10.00 in quarters, and plans to play 100 times or
until she has no quarters left, whichever happens first. Each time
you play a quarter slot machine, you have a 90% chance of losing a quarter.
You have a 9% chance of getting your quarter back, plus $1.00 in quarters.
You also have a 1% chance of getting your quarter back, plus $10.00 in
quarters.
(a) Estimate the expected (average) amount of money Jane has left after
her gambling spree.
(a) Estimate the probability that Jane will have no money left at the
end of her spree.
(c) Estimate the probability that Jane will leave the Casino with more
money than she started with.