Volume 8 (2007), Issue 3, Article 83, 6 pp.
A NOTE ON THE UPPER BOUNDS FOR THE DISPERSION
YU MIAO AND GUANGYU YANG
C OLLEGE OF M ATHEMATICS AND I NFORMATION S CIENCE
H ENAN N ORMAL U NIVERSITY
453007 H ENAN , C HINA .
yumiao728@yahoo.com.cn
D EPARTMENT OF M ATHEMATICS
Z HENGZHOU U NIVERSITY
450052 H ENAN , C HINA .
study_yang@yahoo.com.cn
Received 08 January, 2007; accepted 24 July, 2007
Communicated by N.S. Barnett
A BSTRACT. In this note we provide upper bounds for the standard deviation (σ(X)) , for the
quantity σ 2 (X) + (x − E(X))2 and for the Lp absolute deviation of a random variable. These
improve and extend current results in the literature.
Key words and phrases: Dispersion.
2000 Mathematics Subject Classification. 60E15, 26D15.
1. I NTRODUCTION
Let f : [a, b] ⊂ R → R+ be the probability density function (p.d.f.) of the random variable
X and E(X) and σ(X) the mean and standard deviatin respectively.
In [2], the authors gave upper bounds for the dispersion σ(X) and for σ 2 (X) + (x − E(X))2
in terms of the p.d.f. for a continuous random variable. Recently, Agbeko [1] also obtained
some upper bounds for the same two quantities, namely,
σ(X) ≤ min{max{|a|, |b|}, (b − a)},
(1.1)
and
(1.2)
p
σ 2 (X) + (x − E(X))2 ≤ 2 min{max{|a|, |b|}, (b − a)},
for all x ∈ [a, b].
Both the work of Barnett et al. [2] and Agbeko [1] exclude the discrete case. In this present
note, we remove this exclusion and improve the bounds of (1.1) and (1.2) in Section 2. In
Section 3, we consider the Lp absolute deviation. The symbolism 1A denotes the indicator
function on the set A.
The authors wish to thank the referee for his suggestions in improving the presentation of these results.
010-07
2
Y U M IAO AND G UANGYU YANG
2. S TANDARD D EVIATION
Let P be the distribution of the random variable X, then the expectation of X can be denoted
by
b
Z
E(X) =
xdP(x)
a
and the dispersion or standard deviation of X by
Z b
2
Z b
2
σ(X) =
x dP(x) −
xdP(x) .
a
a
Theorem 2.1. Let X be a random variable with a ≤ X ≤ b. Then we have
σ(X) ≤ min{max{|a|, |b|}, M (a, b, E(X))(b − a)},
(2.1)
where
1
M (a, b, E(X)) = √
2
s
2(b − E(X))2
2(E(X) − a)2
2 − exp −
− exp −
.
(b − a)2
(b − a)2
Proof. Since σ 2 (X) = E(X − E(X))2 , by the Fubini’s theorem, we have
Z (X−E(X))2
2
(2.2)
E(X − E(X)) = E
ds
Z0 ∞
=E
1{(X−E(X))2 ≥s} ds
0
Z ∞
P (X − E(X))2 ≥ s ds
=
0
(b−E(X))2
Z
P(X − E(X) ≥
=
√
s)ds
0
Z
(E(X)−a)2
P −(X − E(X)) ≥
+
√ s ds.
0
For any λ > 0, we have
λ(X−E(X))
Ee
b − E(X) λa E(X) − a λb
≤e
e +
e
b−a
b−a
E(X) − a E(X) − a λ(b−a)
λ(a−E(X))
=e
1−
+
e
b−a
b−a
−λE(X)
, eL(λ) ,
where
E(X) − a E(X) − a λ(b−a)
L(λ) = λ(a − E(X)) + log 1 −
+
e
.
b−a
b−a
The first two derivatives of L(λ) are
(E(X) − a)eλ(b−a)
0
L (λ) = a − E(X) + = a − E(X) + J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 83, 6 pp.
1−
E(X)−a
b−a
+
E(X)−a λ(b−a)
e
b−a
E(X) − a
(1 −
E(X)−a −λ(b−a)
)e
b−a
+
E(X)−a
b−a
,
http://jipam.vu.edu.au/
A
3
NOTE ON THE UPPER BOUNDS FOR THE DISPERSION
00
L (λ) = (b − E(X))(E(X) − a)e−λ(b−a)
2 .
E(X)−a −λ(b−a)
E(X)−a
(1 − b−a )e
+ b−a
Noting that
00
L (λ) ≤
(b − a)2
,
4
then by Taylor’s formula, we have
0
L(λ) ≤ L(0) + L (0)λ +
(b − a)2 2 (b − a)2 2
λ =
λ.
8
8
Therefore from Markov’s inequality, we have
√
√ P X − E(X) ≥ s ≤ inf e− sλ Eeλ(X−E(X))
(2.3)
λ>0
√
(b − a)2 2
λ
≤ inf exp − sλ +
λ>0
8
2s
= exp −
.
(b − a)2
Similarly,
√
2s
P(−(X − E(X)) ≥ s) ≤ exp −
.
(b − a)2
From (2.2), it follows that
(2.4)
E(X − E(X))2
Z (b−E(X))2
Z
√ =
P (X − E(X)) ≥ s ds +
0
(b−E(X))2
P −(X − E(X)) ≥
√ s ds
0
(E(X)−a)2
2s
2s
≤
exp −
ds +
exp −
ds
(b − a)2
(b − a)2
0
0
2(b − E(X))2
2(E(X) − a)2
(b − a)2
=
2 − exp −
− exp −
.
2
(b − a)2
(b − a)2
Furthermore,
Z
(E(X)−a)2
Z
E(X − E(X))2 = E(X 2 ) − (E(X))2 ≤ E(X 2 ) ≤ max{|a|2 , |b2 |},
which, by (2.4), implies the result.
Remark 2.2. In fact, by,
s
2(b − E(X))2
2(E(X) − a)2
1
M (a, b, E(X)) = √
2 − exp −
− exp −
(b − a)2
(b − a)2
2
1 √
≤ √ 2 − e−2 < 1,
2
min{max{|a|, |b|}, M (a, b, E(X))(b − a)} ≤ min{max{|a|, |b|}, (b − a)},
and is, therefore, a tighter bound than (1.1). If E(X) = (b + a)/2, then
s
1
2(b − E(X))2
2(E(X) − a)2
M (a, b, E(X)) = √
2 − exp −
− exp −
(b − a)2
(b − a)2
2
p
1
= 1 − e−1/2 < √ ,
2
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 83, 6 pp.
http://jipam.vu.edu.au/
4
Y U M IAO AND G UANGYU YANG
which means that under some assumptions, this bound is better than the result of [2] for the case
f ∈ L1 ([a, b]).
Corollary 2.3. Let X be a random variable with a ≤ X ≤ b, then for any x ∈ [a, b],
p
(2.5)
σ 2 (X) + (x − E(X))2 ≤ min{2 max{|a|, |b|}, N (a, b, E(X))(b − a)},
where
1
N (a, b, E(X)) = 2 −
2
2
2(b − E(X))2
exp −
(b − a)2
2(E(X) − a)2
+ exp −
.
(b − a)2
Proof. It is clear that (x − E(X))2 ≤ (b − a)2 and from the proof of Theorem 2.1, we know that
1
2(b − E(X))2
2(E(X) − a)2
2
2
σ (X)+(x−E(X)) ≤ 2 −
exp −
+ exp −
(b−a)2 .
2
(b − a)2
(b − a)2
Furthermore,
σ 2 (X) + (x − E(X))2 = E(X − x)2 ≤ max{(x − y)2 , x, y ∈ [a, b]}.
The remainder of the proof follows as Theorem 1.2 in Agbeko [1], giving
max{|x − y|, x, y ∈ [a, b]} ≤ 2 min{max{|a|, |b|}, (b − a)}.
3. Lp A BSOLUTE D EVIATION
In fact by the method in Section 2, we could extend the case of Lp absolute deviation, i.e.,
the following quantity,
σp (X) = E (|X − E(X)|p )1/p .
We have the following
Theorem 3.1. Let X be a random variable and p ≥ 1. Assume that E|X|p < ∞, then its Lp
absolute deviation has the following estimation for p > 2,
(σp (X))p ≤ min{|b − a|p , M1 (a, b, p, X)},
where
(3.1)
(b − a)2
2(|b − E(X)|p ∧ 1)
M1 (a, b, p, X) =
1 − exp −
2
(b − a)2
+ (|b − E(X)|p − |b − E(X)|p ∧ 1)
2(|b − E(X)|p ∧ 1)2/p
× exp −
(b − a)2
(b − a)2
2(|a − E(X)|p ∧ 1)
+
1 − exp −
2
(b − a)2
p
p
+ (|a − E(X)| − |a − E(X)| ∧ 1)
2(|a − E(X)|p ∧ 1)2/p
× exp −
.
(b − a)2
If 1 ≤ p < 2, then we have
(σp (X))p ≤ min{|b − a|p , M2 (a, b, p, X)},
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 83, 6 pp.
http://jipam.vu.edu.au/
A
5
NOTE ON THE UPPER BOUNDS FOR THE DISPERSION
where
(3.2) M2 (a, b, p, X)
(b − a)2
2|b − E(X)|p ∧ 1
2|b − E(X)|p
= |b − E(X)| ∧ 1 +
exp −
− exp −
2
(b − a)2
(b − a)2
2|a − E(X)|p
(b − a)2
2|a − E(X)|p ∧ 1
p
+ |a − E(X)| ∧ 1 +
exp −
− exp −
.
2
(b − a)2
(b − a)2
p
Proof. On one hand, by the Fubini’s theorem, we have
(3.3)
Z
p
|X−E(X)|p
E|X − E(X)| = E
ds
0
Z
=E
Z
=
∞
1{|X−E(X)|p ≥s} ds
0
∞
P(|X − E(X)|p ≥ s)ds
0
|b−E(X)|p
Z
P (X − E(X)) ≥ s1/p ds
=
0
Z
|E(X)−a|p
P −(X − E(X)) ≥ s1/p ds.
+
0
From the estimation (2.3), we have
Z |b−E(X)|p
Z |E(X)−a|p
p
1/p
E|X − E(X)| =
P(X − E(X) ≥ s )ds +
P(−(X − E(X)) ≥ s1/p )ds
0
0
Z |b−E(X)|p
Z |E(X)−a|p
2/p
2s2/p
2s
≤
exp −
ds +
exp −
ds.
(b − a)2
(b − a)2
0
0
(Case: p > 2.) By simple calculating, we have
Z |b−E(X)|p
2s2/p
exp −
ds
(b − a)2
0
Z |b−E(X)|p ∧1
Z |b−E(X)|p
2s
2s2/p
≤
exp −
ds +
exp −
ds
(b − a)2
(b − a)2
0
|b−E(X)|p ∧1
(b − a)2
2(|b − E(X)|p ∧ 1)
≤
1 − exp −
2
(b − a)2
2(|b − E(X)|p ∧ 1)2/p
p
p
+ (|b − E(X)| − |b − E(X)| ∧ 1) exp −
(b − a)2
and with the same reason
Z
0
|E(X)−a|p
2s2/p
exp −
(b − a)2
2(|a − E(X)|p ∧ 1)
1 − exp −
(b − a)2
2(|a − E(X)|p ∧ 1)2/p
p
p
+ (|a − E(X)| − |a − E(X)| ∧ 1) exp −
.
(b − a)2
(b − a)2
ds ≤
2
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 83, 6 pp.
http://jipam.vu.edu.au/
6
Y U M IAO AND G UANGYU YANG
(Case 1 ≤ p < 2) In this case, we have
Z |b−E(X)|p
2s2/p
exp −
ds
(b − a)2
0
Z |b−E(X)|p ∧1
Z |b−E(X)|p
2s2/p
2s
≤
exp −
ds +
exp −
ds
(b − a)2
(b − a)2
0
|b−E(X)|p ∧1
2|b − E(X)|p ∧ 1
2|b − E(X)|p
(b − a)2
p
≤ |b − E(X)| ∧ 1 +
exp −
− exp −
2
(b − a)2
(b − a)2
and with the same reason
Z |E(X)−a|p
2s2/p
exp −
ds ≤ |a − E(X)|p ∧ 1
2
(b
−
a)
0
2|a − E(X)|p ∧ 1
2|a − E(X)|p
(b − a)2
exp −
− exp −
.
+
2
(b − a)2
(b − a)2
On the other hand, for any p ≥ 1, it is clear that
E|X − E(X)|p ≤ (b − a)p .
From the above discussion, the desired results are obtained.
Remark 3.2. M1 (a, b, p, X) and M2 (a, b, p, X) are sometimes better than (b−a)p , e.g., if p > 2,
taking E(X) = (a + b)/2 and letting 1/2 < (b − a)/2 < 1, then
M1 (a, b, p, X) = (b − a)2 1 − exp −21−p (b − a)p−2
< (b − a)p 1 − exp −21−p (b − a)p−2
< (b − a)p .
Further, if 1 ≤ p < 2, taking E(X) = (a + b)/2 and letting (b − a)/2 < 1, then
(b − a)p
M2 (a, b, p, X) = 2
= 21−p (b − a)p ≤ (b − a)p .
2p
R EFERENCES
[1] N.K. AGBEKO, Some p.d.f.-free upper bounds for the dispersion σ(X) and the quantity σ 2 (X) +
(x−E(X))2 , J. Inequal. Pure and Appl. Math., 7(5) (2006), Art. 186. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=803].
[2] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, Some inequalities for
the dispersion of a random variable whose pdf is defined on a finite interval, J. Inequal. Pure and
Appl. Math., 2(1) (2001), Art. 1. [ONLINE: http://jipam.vu.edu.au/article.php?
sid=117].
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 83, 6 pp.
http://jipam.vu.edu.au/