01:960:285 Lecture Notes - Lecture 2: Bayes Estimator, Sampling Distribution, Rivet

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Rutgers University - Stat 285, Section 07 Spring 2017
Second MIDTERM March 30, 2017
SOLUTIONS
BAYES Problem 1:
A total of 46 percent of the voters in a certain city classify themselves as Independents,
whereas 30 percent classify themselves as Liberals and 24 percent say that they are
Conservatives. In a recent local election, 35 percent of the Independents, 62
percent of the Liberals, and 58 percent of the Conservatives voted. A voter is chosen at
random. Given that this person voted in the local election, what is the probability that he
or she is an Independent? (15 points)
Solution Problem 1:
Let I, L, and C be the event that a random person is an independent, liberal, or a
conservative respectively. Let V be the event that a person voted. Then from the
problem we are given that
P(I) = 0.46 , P(L) = 0.3 , P(C) = 0.24
and
P(V |I) = 0.35 , P(V |L) = 0.62 , P(V |C) = 0.58 .
We want to compute P(I|V ), which by Baye’s rule is given by;
P(I|V ) =
)()|)()|()()|(
)()|(
)(
)()|(
CPCPVLPLVPIPIVP
IPIVP
VP
IPIVP
++
=
.
We need to calculate P(V ) which we do (as above) by conditioning on the various types
of voters: P(V ) = P(V |I)P(I) + P(V |L)P(L) + P(V |C)P(C)
= 0.35(0.46) + 0.62(0.3) + 0.58(0.24) = 0.4862 .
Then the requested conditional probability is given by
P(I|V ) =
3311.0
4862.0
)46.0(35.0 =
Binomial Problem 2
An airline, having observed that 2% of the people making reservations fail to show up
for the flight, sells 100 tickets for a flight that has 98 seats. (That is, they assume that at
least 2 people will fail to show up for the flight.)
What is the probability that fewer than 2 people fail to show up for the flight? [Assume
that the people act independently, i.e., assume that each person shows upor does not
show upindependently of the behavior of the other people.] Give a mathematical
expression for the exact answer.
[Note: You do not necessarily have to compute an exact decimal number, but your final
answer should have only numbers in it (i.e. no parameters such as n, p or x) and use
only the operations of multiplication, addition and raising to a power.] (14 points)
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