We suppose, without repeated assertion, that the random variables and
functions of random vectors are integrable, as needed.
- (CE1): Defining condition. e(X)=E[g(Y)|X]e(X)=E[g(Y)|X] a.s. iff
E[IM(X)g(Y)]=E[IM(X)e(X)]E[IM(X)g(Y)]=E[IM(X)e(X)]
for each Borel set M on the codomain of X.
- (CE1a): If P(X∈M)>0P(X∈M)>0, then E[IM(X)e(X)]=E[g(Y)|X∈M]P(X∈M)E[IM(X)e(X)]=E[g(Y)|X∈M]P(X∈M)
- (CE1b): Law of total probability. E[g(Y)]=E{E[g(Y)|X]}E[g(Y)]=E{E[g(Y)|X]}
- (CE2): Linearity. For any constants a,ba,b
E
[
a
g
(
Y
)
+
b
h
(
Z
)
|
X
]
=
a
E
[
g
(
Y
)
|
X
]
+
b
E
[
h
(
Z
)
|
X
]
a
.
s
.
E
[
a
g
(
Y
)
+
b
h
(
Z
)
|
X
]
=
a
E
[
g
(
Y
)
|
X
]
+
b
E
[
h
(
Z
)
|
X
]
a
.
s
.
(Extends to any finite linear combination) - (CE3): Positivity; monotonicity.
- g(Y)≥0g(Y)≥0 a.s. implies E[g(Y)|X]≥0E[g(Y)|X]≥0 a.s.
- g(Y)≥h(Z)g(Y)≥h(Z) a.s. implies E[g(Y)|X]≥E[h(Z)|X]E[g(Y)|X]≥E[h(Z)|X] a.s.
- (CE4): Monotone convergence. Yn→YYn→Y a.s. monotonically implies
E
[
Y
n
|
X
]
→
E
[
Y
|
X
]
a
.
s
.
E
[
Y
n
|
X
]
→
E
[
Y
|
X
]
a
.
s
.
- (CE5): Independence. {X,Y}{X,Y} is an independent pair
- iff E[g(Y)|X]=E[g(Y)]E[g(Y)|X]=E[g(Y)] a.s. for all Borel functions g
- iff E[IN(Y)|X]=E[IN(Y)]E[IN(Y)|X]=E[IN(Y)] a.s. for all Borel sets N
on the codomain of Y
- (CE6): e(X)=E[g(Y)|X]e(X)=E[g(Y)|X] a.s. iff
E
[
h
(
X
)
g
(
Y
)
]
=
E
[
h
(
X
)
e
(
X
)
]
a
.
s
.
for
any
Borel
function
h
E
[
h
(
X
)
g
(
Y
)
]
=
E
[
h
(
X
)
e
(
X
)
]
a
.
s
.
for
any
Borel
function
h
- (CE7): E[h(X)|X]=h(X)E[h(X)|X]=h(X) a.s. for any Borel function h
- (CE8): E[h(X)g(Y)|X]=h(X)E[g(Y)|X]E[h(X)g(Y)|X]=h(X)E[g(Y)|X] a.s. for any Borel function h
- (CE9): If X=h(W)X=h(W), then E{E[g(Y)|X]|W}=E{E[g(Y)|W]|X}=E[g(Y)|X]E{E[g(Y)|X]|W}=E{E[g(Y)|W]|X}=E[g(Y)|X], a.s.
- (CE9a): E{E[g(Y)|X]|X,Z}=E{E[g(Y)|X,Z]|X}=E[g(Y)|X]E{E[g(Y)|X]|X,Z}=E{E[g(Y)|X,Z]|X}=E[g(Y)|X] a.s.
- (CE9b): If X=h(W)X=h(W) and W=k(X)W=k(X), with h,kh,k Borel functions,
then
E
[
g
(
Y
)
|
X
]
=
E
[
g
(
Y
)
|
W
]
a
.
s
.
E
[
g
(
Y
)
|
X
]
=
E
[
g
(
Y
)
|
W
]
a
.
s
.
- (CE10): If g is a Borel function such that E[g(t,Y)]E[g(t,Y)] is finite for all t on
the range of X and E[g(X,Y)]E[g(X,Y)] is finite, then
- E[g(X,Y)|X=t]=E[g(t,Y)|X=t]E[g(X,Y)|X=t]=E[g(t,Y)|X=t] a.s. [PX][PX]
- If {X,Y}{X,Y} is independent, then E[g(X,Y)|X=t]=E[g(t,Y)]E[g(X,Y)|X=t]=E[g(t,Y)] a.s. [PX][PX]
- (CE11): Suppose {X(t):t∈T}{X(t):t∈T} is a real-valued, measurable random
process whose parameter set T is a
Borel subset of the real line and S
is a random variable whose range is a subset of T, so that
X(S)X(S) is a random variable.
If E[X(t)]E[X(t)] is finite for all t in T and
E[X(S)]E[X(S)] is finite, then
- E[X(S)|S=t]=E[X(t)|S=t]E[X(S)|S=t]=E[X(t)|S=t] a.s. [PS][PS]
- If, in addition, {S,XT}{S,XT} is independent, then
E[X(S)|S=t]=E[X(t)]E[X(S)|S=t]=E[X(t)] a.s. [PS][PS]
- (CE12): Countable additivity and countable sums.
- If Y is integrable on A and A=⋁n=1∞AnA=⋁n=1∞An,
then E[IAY|X]=∑n=1∞E[IAnY|X]E[IAY|X]=∑n=1∞E[IAnY|X] a.s. - If ∑n=1∞E[|Yn|]<∞∑n=1∞E[|Yn|]<∞ , then
E∑n=1∞Yn|X=∑n=1∞E[Yn|X]E∑n=1∞Yn|X=∑n=1∞E[Yn|X] a.s.
- (CE13): Triangle inequality. |E[g(Y)|X]|≤E[|g(Y)||X]|E[g(Y)|X]|≤E[|g(Y)||X] a.s.
- (CE14): Jensen's inequality. If g is a convex function on an interval I
which contains the range of a real random variable Y, then
g
{
E
[
Y
|
X
]
}
≤
E
[
g
(
Y
)
|
X
]
a
.
s
.
g
{
E
[
Y
|
X
]
}
≤
E
[
g
(
Y
)
|
X
]
a
.
s
.
- (CE15): Suppose E[|Y|p]<∞E[|Y|p]<∞ and E[|Z|p]<∞E[|Z|p]<∞ for
1≤p<∞1≤p<∞. Then
E
{
|
E
[
Y
|
X
]
-
E
[
Z
|
X
]
|
p
}
≤
E
[
|
Y
-
Z
|
p
]
<
∞
E
{
|
E
[
Y
|
X
]
-
E
[
Z
|
X
]
|
p
}
≤
E
[
|
Y
-
Z
|
p
]
<
∞
