Econ. 240A, Spring 1998 D. McFadden
PROBLEM SET 1 (Review of Elementary Probability theory)
(Due Monday, Feb. 2, with discussion in section on Feb. 4)
1. Prove that a σ-field of events contains
countable
intersections of its members.
2. It is known that 0.2 percent of the population is HIV-positive. It is known that
a screening test for HIV has a 10 percent chance of incorrectly showing positive if
you are not, and a 2 percent chance of incorrectly showing negative if you are in
truth positive. What proportion of the population that tests positive is in truth
positive?
3. Ramanathan, problems 2.4, 2.5, 2.7.
4. An airplane has 40 seats. The probability that a ticketed passenger shows up for
the flight is 0.95, and the events that any two different passengers show up is
statistically independent. If the airline sells 45 seats, what is the probability
that the plane will be overbooked? How many seats can the airline sell, and keep the
probability of overbooking to 5 percent or less?
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5. Prove that the expectation E(X - c) is minimized when c = EX.
6. Prove that the expectation EX-cis minimized when c = median(X).
7. A sealed bid auction has an economist for sale to the highest of n bidders. You
are bidder 1. Your experience is that the bids of each other bidder is distributed
α
with a Power distribution F(X) = X for 0 ≤X≤1. Your profit if you are successful
in buying the economist at price y is 1 - y. What should you bid to maximize your
expected profit?
8. A random variable X has a normal distribution if its density is
f
(x) =
22
1 -(x-µ)/2σ2
-------------------------
⋅e , where µand σare parameters. Prove that X has mean µand
q=======
6
e
2π⋅σ 2344
variance σ. Prove that E(X-µ) = 0 and E(X-µ)=3σ.
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