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Economic Statistics Exam 2
Last name:
November 2, 2006
First name:
PPID:
Circle class time:
9:30

11:00
12:30

2:00
Please sign that you abide by the honor pledge:
This exam contains
fifteen
short answer questions
(
worth 4 points each
)
and
four
long answer questions
(
worth 15 points each
)
. Please answer in the space provided; if
you need more room, indicate clearly that the answer is continued on the back of the
page. In all problems where you need to make a calculation, simplify your answer as
much as possible.
Show your work clearly to be eligible for partial credit.
Some PDFs:
Some probabilities:
f
(
x
)
=
1
2
!"
2
e
#
(
X
#
μ
)
2
"
2
P
[
x
!
a
]
=
e
"
#
a
f
(
x
)
=
1
u
!
!
P
[
X
]
=
x
e
!
X
!
f
(
x
)
=
!
e
"
x
P
[
X
]
=
N
!
X
!(
N
!
X
)!
p
X
(1
!
p
)
N
!
X
Short Questions
(
4 points apiece
)
1.
X
is a random variable with a continuous distribution. Its probability density
function is some function,
f
(
x
)
. How would we find the probability that
X
is
between two values
(
for example, between 2.13 and 4.56
)
, using the PDF?
P
[2.13
!
X
!
4.56]
=
f
(
x
)
dx
2.13
4.56
"
2.
The sum of many small random variables tends to have a normal distribution,
regardless of the distribution of the individual random variables. What is the
analogous statement for the
log

normal
distribution?
The
product
of many small random variables tends to have a
lognormal
distribution.
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View Full DocumentEcon 400 Midterm 2, page 2 of 8.
3.
What is the difference between an
estimate
and an
estimator?
The estimator is a rule or formula used to
“
guess
”
the value of some statistic; an estimate is the
value you obtain with a specific sample.
4.
After a long night of trick

or

treating, the number of candies in my bag is a
random variable, distributed normally with mean of 300 and variance of 400. What
is the chance that I have fewer than 250 candies in my bag?
The z

score is
250
!
300
400
=
!
50
20
=
!
2.5
.
P
[
X
<
250]
=
F
z
(
!
2.5)
=
0.0062.
5.
The lifespan of a male born in the U.S. in 1999 is 74 years. What is the chance
that a person lives to be this age or older?
(
You must decide the most appropriate
distribution to use in this case.
)
Durations are usually modeled using the exponential distribution. Given the exponential
distribution, the probability of living to be 74 or older is
P
[
x
!
74]
=
e
"
#
74
, where
!
=
1/
μ
=
1/ 74
. Thus,
P
[
x
!
74]
=
e
"
74/74
=
e
"
1
.
6.
State the
central limit theorem.
As the sample size
(
N
)
gets large, the sample mean
(
X
)
has an approximately normal
distribution with mean
X
=
X
and variance
X
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