Econ 140
Fall 2011
Economics 140
September 13, 2011
1
Homework #1
Due September 21, 2011 at beginning of lecture
Chapter 2: 4, 6, C7
More problems will be added before the due date.
Please be sure to submit either a do or
log file of your Stata work with your problem set. Only hard copies will be accepted; no emailed
assignments, please.
2
Some more math
Some properties of expectations:
Expectation of functions:
E
[
g
(
Y
)] =
X
y
∈
Y
f
(
y
)
g
(
y
) (discrete)
E
[
g
(
Y
)] =
Z
∞
∞
f
(
y
)
g
(
y
)
dy
(continuous)
Expectation of a function is not equal to the function of the expectation:
E
[
g
(
y
)]
6
=
g
(
E
[
Y
])
Linearity:
E
[
ag
(
Y
) +
bh
(
Y
) +
c
] =
aE
[
g
(
Y
)] +
bE
[
h
(
Y
)] +
E
[
c
] =
aE
[
g
(
Y
)] +
bE
[
h
(
Y
)] +
c
E
[
X
+
Y
] =
E
[
X
] +
E
[
Y
]
E
[
N
X
i
=1
A
i
] =
N
X
i
=1
E
[
A
i
]
Law of Iterated Expectations
:
E
[
Y
] =
X
X
x
=1
E
[
Y

X
]
Pr
[
X
=
x
] (discrete)
E
[
Y
] =
Z
∞
∞
E
[
Y

X
]
f
(
x
) (continuous)
Suppose we are trying to calculate the expected value of weight (Y), and we have information
about gender (X) as well. In this case, we can use the law of iterated expectations to use information
about average weight, given gender, to estimate the average weight of a person:
E
[
Y
] =
2
X
x
=1
E
[
Y

X
]
Pr
[
X
=
x
]
=
E
[
weight

male
]
Pr
[
X
=
male
] +
E
[
weight

female
]
Pr
[
X
=
female
]
1
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Econ 140
Fall 2011
3
Some definitions
Recall how we can characterize the distribution of a random variable with moments:
1. The
expected value (mean)
is the value we expect the random variable to take in the
population, or the measure of central tendency.
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 Spring '08
 DUNCAN
 Economics, Standard Deviation, Variance, Probability theory

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