Lecture 5: Inequalities
51
Lecture 5 : Inequalities
5.7 Inequalities
Let
X,Y
etc. be real r.v.’s deﬁned on (Ω
,
F
,
P
).
Theorem 5.7.1 (Jensen’s Inequality)
Let
ϕ
be convex,
E
(

X

)
<
∞
,
E
(

ϕ
(
X
)

)
<
∞
. Then
ϕ
(
E
(
X
))
≤
E
(
ϕ
(
X
))
(5.11)
Proof Sketch:
As
ϕ
is convex,
ϕ
is the supremum of a countable collection of lines.
ϕ
(
x
) = sup
n
L
n
(
x
)
,
L
n
(
x
) =
a
n
x
+
b
n
L
n
(
E
X
)
(1)
=
E
(
L
n
(
X
))
(2)
≤
E
(
ϕ
(
X
))
Take sup on
n
.
(1) used linearity, (2) used monotonicity.
Keep the following example in mind to remember the direction of the inequality.
Example 5.7.2
E
X
2
≥
(
E
X
)
2
.
(5.12)
In other words, if we deﬁne Var
(
X
) =
E
(
X

E
(
X
))
2
then we get Var
(
X
)
≥
0
.
Example 5.7.3
Let
Ω =
{
1
,
1
}
n
and
S
n
(
x
) =
∑
n
i
=1
x
i
then we can calculate
E
(
S
n
(
x
)) =
E
±
1
2
(
2
1
·
2
S
n

1
(
x
)
+ 2

1
·
2
S
n

1
(
x
)
)
¶
=
5
4
E
(2
S
n

1
(
x
)
)
.
By induction we see that
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 Spring '09
 Reed
 Inequalities, Order theory, Monotonic function, Convex function

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