0.1
Exercises
1. Let
V ar
(
X
F
) =
E
(
X
2
F
)

(
E
(
X
F
))
2
. Show that
V ar
(
X
) =
E
(
V ar
(
X
F
)) +
V ar
(
E
(
X
F
))
.
Proof.
RHS
=
E
(
V ar
(
X
F
)) +
V ar
(
E
(
X
F
))
=
‡
E
‡
E
(
X
2
F
)

(
E
(
X
F
))
2
··
+
‡
E
(
E
(
X
F
))
2

(
EE
(
X
F
))
2
·
=
EE
(
X
2
F
)

(
EE
(
X
F
))
2
=
E
(
X
2
)

(
EX
)
2
=
LHS
2. Show that if
X
and
Y
are r.v.’s with
E
(
Y
F
) =
X
and
EX
2
=
EY
2
<
∞
, then
X
=
Y
a.s. (i.e.
P
(
X
=
Y
) = 1).
Proof.
Note that
X
=
E
(
Y
F
)
∈ F
, hence
E
(
X

Y
)
2
=
EX
2
+
EY
2

2
EXY
= 2
EX
2

2
EE
(
XY
F
)
=
2
EX
2

2
E
[
XE
(
Y
F
)] = 2
EX
2

2
EX
2
= 0
,
implying that
P
(
X

Y
= 0) = 0.
3. Let
X
i
be independent with
EX
i
= 0 and
σ
2
i
=
V ar
(
X
i
)
<
∞
, and let
S
2
n
=
∑
n
i
=1
X
2
i
and
B
2
n
=
ES
2
n
=
∑
n
i
=1
σ
2
i
. Show that
S
2
n

B
2
n
is a martingale (w.r.t. the natural
filtration.)
Proof.
Clearly,
E

S
2
n

<
∞
, and
S
2
n
∈ F
n
:=
σ
(
X
1
, ..., X
n
). Also,
E
(
S
2
n
+1
F
n
)
=
S
2
n
+
E
(
X
2
n
+1

σ
n
+1
F
n
)
=
S
2
n
+
E
(
X
2
n
+1

σ
2
n
+1
)
=
S
2
n
.
4. Let
X
i
be nonnegative i.i.d.
r.v.’s with
EX
i
= 1 and
P
(
X
i
= 1)
<
1.
Show that
T
n
=
Q
n
i
=1
X
i
is a martingale (w.r.t. the natural filtration.)
Proof.
Clearly,
E

T
n

=
Q
n
i
=1
EX
i
= 1
<
∞
, and
T
n
∈ F
n
:=
σ
(
X
1
, ..., X
n
). Also,
E
(
T
n
+1
F
n
) =
T
n
E
(
X
n
+1
F
n
) =
T
n
E
(
X
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 Spring '11
 D
 Trigraph, Tn, F

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