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

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23 Dec 2018

School

Department

Course

Professor

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 up—or does not

show up—independently 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)