Otober 3, 2015
4
MATHEMATICAL PRELIMINARIES REDUX
tail inequalities
Markov's inequality
Bienayme
Chebyshev
Jensen's inequality
onvex
onave
rst moment priniple
seond moment priniple
We've already disussed the most imp ortant tehnique of this kind in Se-
tion 1.2.10. Stated in highly general terms, the basi idea an be formulated as
follows:
Let
f
be any nonnegative funtion suh that
f
(
x
)
s >
0
when
x
2
S
.
Then
Pr(
X
2
S
)
E
f
(
X
)
=s;
(
15
)
provided that
Pr(
X
2
S
)
and
E
f
(
X
)
both exist.
For example,
f
(
x
) =
j
x
j
yields
Pr(
j
X
j
m
)
E
j
X
j
=m
(
16
)
whenever
m >
0. The pro of is amazingly simple, beause we obviously have
E
f
(
X
)
Pr(
X
2
S
)
s
+ Pr(
X =
2
S
)
0
:
(
17
)
Formula (
15
) is often alled
Markov's inequality
, b eause A. A. Markov disussed
the sp eial ase
f
(
x
) =
j
x
j
a
in
Izv
est
a Imp. Akad. Nauk
(6)
1
(1907), 707{716.
If we set
f
(
x
) = (
x
E
X
)
2
, we get the famous 19th-entury inequality of
Bienayme and Chebyshev:
Pr
j
X
E
X
j
r
var(
X
)
=r
2
:
(
18
)
The ase
f
(
x
) =
e
ax
is also extremely useful.
Another fundamental estimate, known as
Jensen's inequality
[
Ata Mathe-
matia
30
(1906), 175{193℄, applies to
onvex
funtions
f
; we've seen it so far
only as a \hint" to exerise 6.2.2{36(!). The real-valued funtion
f
is said to b e
onvex in an interval
I
of the real line, and
f
is said to be onave in
I
, if
f
(
px
+
q y
)
pf
(
x
) +
q f
(
y
) for all
x; y
2
I ;
(
19
)
whenever
p
0,
q
0, and
p
+
q
= 1. This ondition turns out to b e equivalent to
saying that
f
00
(
x
)
0 for all
x
2
I
, if
f
has a seond derivative
f
00
. For example,
the funtions
e
ax
and
x
2
n
are onvex for all onstants
a
and all nonnegative
integers
n
; and if we restrit onsideration to p ositive values of
x
, then
f
(
x
) =
x
n
is onvex for
al l
integers
n
(notably
f
(
x
) = 1
=x
when
n
=
1). The funtions
ln(1
=x
) and
x
ln
x
are also onvex for
x >
0. Jensen's inequality states that
f
(E
X
)
E(
f
(
X
)) (
20
)
when
f
is onvex in the interval
I
and the random variable
X
takes values only
in
I
. (See exerise 42 for a pro of.) For example, we have 1
=
E
X
E(1
=X
) and
ln E
X
E ln
X
and (E
X
) ln E
X
E(
X
ln
X
), when
X
is p ositive. Notie that
(
20
) atually redues to the very denition of onvexity, (
19
), in the sp eial ase
when
X
=
x
with probability
p
and
X
=
y
with probability
q
.
Third and fourth on our list of remarkably useful inequalities are two lassial
results that apply to any random variable
X
whose values are nonnegative
integers:
Pr(
X >
0)
E
X
; (\the rst moment priniple") (
21
)
Pr(
X >
0)
(E
X
)
2
=
(E
X
2
)
:
(\the seond moment priniple") (
22
)
Formula (
21
) is obvious, b eause the left side is
p
1
+
p
2
+
p
3
+
when
p
k
is the
probability that
X
=
k
, while the right side is
p
1
+ 2
p
2
+ 3
p
3
+
.
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