AMS 311, Spring Semester, 2010
Chapter Seven
Properties of Expectation
A corollary of Proposition 2.1 is that
],
[
]
[
]
[
Y
E
X
E
Y
X
E
+
=
+
provided that the
expectations are finite. By induction,
],
[
]
[
]
[
]
[
2
1
2
1
n
n
X
E
X
E
X
E
X
X
X
E
+
+
+
=
+
+
provided each expectation is finite.
Example 2c
. Let
n
X
X
X
,
,
,
2
1
be jointly distributed random variables, each with
expected value
.
μ
Prove that
.
]
[
]
[
1
μ
=
=
∑
=
n
X
E
X
E
n
i
i
n
Note that the only assumption
needed is that the random variables have the same finite mean.
Definition
The covariance between
X
and
Y
, denoted by
Cov
(
X
,
Y
), is defined by
])].
[
])(
[
[(
)
,
(
Y
E
Y
X
E
X
E
Y
X
Cov


=
One identity that is often helpful is that
].
[
]
[
]
[
)
,
(
Y
E
X
E
XY
E
Y
X
Cov

=
Proposition 3.2
(i)
).
,
cov(
)
,
cov(
X
Y
Y
X
=
(ii)
).
var(
)
,
cov(
X
X
X
=
(iii)
).
,
cov(
)
,
cov(
X
Y
a
Y
aX
=
(iv)
.
)
,
cov(
)
,
cov(
1
1
1
1
∑∑
∑
∑
=
=
=
=
=
n
i
m
j
j
i
m
j
j
n
i
i
Y
X
Y
X
Example 3a
. Let
n
X
X
X
,
,
,
2
1
be independent and identically distributed random
variables having expected value
μ
and variance
.
2
σ
Let
,
1
n
X
X
n
i
i
n
∑
=
=
and let
.
1
)
(
1
2
2


=
∑
=
n
X
X
S
n
i
n
i
Compute
)
var(
n
X
and
].
[
2
S
E
Definition
The correlation of two random variables
X
and
Y
, denoted by
)
,
(
Y
X
ρ
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 Spring '08
 Tucker,A
 Probability theory, moment generating function

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