AMS 311, Spring Semester, 2010
Chapter Seven
Properties of Expectation
Proposition 2.1
If
X
and
Y
have a joint probability mass function
p(x,y)
, then
,
)
,
(
)
,
(
)]
,
(
[
∑∑
=
y
x
y
x
p
y
x
g
Y
X
g
E
provided that the sum is absolutely convergent.
If
X
and
Y
have a joint probability density function
f(x,y)
, then
,
)
,
(
)
,
(
)]
,
(
[
∫ ∫
∞
∞

∞
∞

=
dxdy
y
x
f
y
x
g
Y
X
g
E
provided that the integral is absolutely convergent.
A corollary of this theorem 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.
In Example 2d, this result is used to study Boole’s
inequality.
Example 2g
.
Mean of a hypergeometric random variable
. If
n
balls are randomly selected
from an urn containing
N
balls of which
m
are white, find the expected number of white
balls selected. Answer is
.
N
mn
Example 2q
. Let
X
be a nonnegative, integervalued random variable. Prove that
.
}
[
]
[
1
∑
=
≥
=
n
i
i
X
P
X
E
CauchySchwartz inequality generalizes to expectations:
.
]
[
]
[
]
[
2
2
Y
E
X
E
XY
E
≤
Example
There are three random variables
X
,
Y
, and
Z
with
v
a r (
)
v
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 Spring '08
 Tucker,A
 Probability theory, probability density function, Probability mass function, joint probability density, conditional variance formula

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