Statistics Graduate Level Quiz
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PROBABILITY FOUNDATIONS
Question 1
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PASSAGE
Let
X
X
X
be integrable, and let sigma-fields
H
H
H
and
G
G
G
satisfy
H
⊆
G
H⊆G
H
⊆
G
. Let
Y
Y
Y
be a bounded,
H
H
H
-measurable random variable.
Which statement about
E
[
Y
{
E
(
X
∣
G
)
−
E
(
X
∣
H
)
}
]
E[Y\{E(X\mid G)-E(X\mid H)\}]
E
[
Y
{
E
(
X
∣
G
)
−
E
(
X
∣
H
)}]
is correct?
A
It equals
0
0
0
under the stated assumptions.
B
It equals
E
(
Y
X
)
−
E
(
Y
)
E
(
X
)
E(YX)-E(Y)E(X)
E
(
Y
X
)
−
E
(
Y
)
E
(
X
)
under the stated assumptions.
C
It equals
0
0
0
only when
X
X
X
is independent of
G
G
G
.
D
It equals
0
0
0
only when
Y
Y
Y
is constant almost surely.
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