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Useful Properties
Econ 410  Fall 2006. TA: Ignacio Monzon
1
Properties of Expectations
Assume
X
and
Y
are random variables and
a
and
b
are given constants.
Property 1
The expected value of a constant is the same constant:
E
(
a
) =
a
Property 2
The expected value is a linear operator:
E
(
aX
+
b
) =
aE
(
X
) +
b
Property 3
E
(
aX
+
bY
) =
aE
(
X
) +
bE
(
Y
)
Things that are NOT TRUE in general
It is NOT true that
E
X
Y
±
=
E
(
X
)
E
(
Y
)
it is NOT true that
E
(
X
2
) =
E
(
X
)
2
Properties of Variances
Property 1
V ar
(
X
) = 0
if and only if
X
is a constant (and then, not a random variable)
Property 2
V ar
(
aX
+
b
) =
a
2
V ar
(
X
)
Property 3
V ar
(
aX
+
bY
) =
a
2
V ar
(
X
) +
b
2
V ar
(
Y
) + 2
abCov
(
X;Y
)
:
Properties of Conditional Expectations
Assume
X
and
Y
are random variables and that
g
(
±
)
and
h
(
±
)
are two functions.
Property 1
E
[
g
(
X
)
j
X
] =
g
(
X
)
Property 2
E
[
g
(
X
)
Y
+
h
(
X
)
j
X
] =
g
(
X
)
E
(
Y
j
X
) +
h
(
X
)
Property 3
If
X
and
Y
are independent, then
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This note was uploaded on 07/28/2010 for the course ECON 410 taught by Professor Staff during the Fall '08 term at Wisconsin.
 Fall '08
 Staff
 Econometrics

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