Ms. Cooper wishes to attend a conference which would take place in Hotel Mariott or Hotel Hilton. She is
not sure where the conference will take place but has to make her decision to book a room, immediately.
The rooms in Hotel Mariott cost $180 per night and those in Hilton cost $150 per night. The taxi charge
from Hotel Mariott to Hotel Hilton and vice-versa is $30. The odds that the conference will be held in Hotel
Mariott is 5 : 1. The conference is a two-day event. So she needs to book a room for two nights. If she stays
in the wrong hotel, she needs to take a taxi from the hotel she is staying to the conference venue and back
on day 1 and again book a taxi once on day 2. Assume that the taxi charges to the airport from both hotels
are same, $60.
a) (5 points) Which hotel should she book a room in, if she wants to minimize her risk or expected cost?
Explain your solution in detail.
b) (10 points) If she decides which hotel she wants to reserve by flipping a fair coin, with Pr{heads} =
Pr{tails} = 1
2 , then show the risk set and the admissible rules.
4) (5 points) Consider a random variable, U(0, 1). It is desired to guess the value of by making three
observations of a random variable, X1, X2 and X3, where Xi Bernoulli( ), i = 1, 2, 3. Is S = X1+2X2+3X3
6
a sufficient statistic for ? Justify your answer (Hint: See if you can come up with two or more different
combinations of X1, X2 and X3, that yield the same value of S).
5) (5 points) A communication system transmits 0’s or 1’s with equal probability.
(10 points) Your company must make a sealed bid for a construction project. If you succeed in winning
the contract (by having the lowest bid), then you plan to pay another firm $100, 000 to do the work. If you
believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled
as the value of a random variable that is uniformly distributed on (70, 140), how much should you bid to
maximize your expected profit? Explain all the steps in your solution in detail (Hint: If you want to win as
well as make profit you need to bid at least $100, 000).
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