A NOTE ON THE UPPER BOUNDS FOR THE
DISPERSION
Upper Bounds for the
Dispersion
YU MIAO
GUANGYU YANG
College of Mathematics and Information Science
Henan Normal University
453007 Henan, China.
EMail: yumiao728@yahoo.com.cn
Department of Mathematics
Zhengzhou University
450052 Henan, China.
EMail: study_yang@yahoo.com.cn
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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Received:
08 January, 2007
Accepted:
24 July, 2007
Communicated by:
JJ
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N.S. Barnett
J
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2000 AMS Sub. Class.:
60E15, 26D15.
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Key words:
Dispersion.
Abstract:
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.
Acknowledgements:
The authors wish to thank the referee for his suggestions in improving the presentation
of these results.
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Contents
1
Introduction
3
2
Standard Deviation
4
3
Lp Absolute Deviation
9
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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1.
Introduction
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)
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
and
(1.2)
Upper Bounds for the
Dispersion
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.
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2.
Standard Deviation
Let P be the distribution of the random variable X, then the expectation of X can be
denoted by
Z
b
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) .
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
a
a
Theorem 2.1. Let X be a random variable with a ≤ X ≤ b. Then we have
(2.1)
Upper Bounds for the
Dispersion
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σ(X) ≤ min{max{|a|, |b|}, M (a, b, E(X))(b − a)},
where
1
M (a, b, E(X)) = √
2
s
2(b − E(X))2
2 − exp −
(b − a)2
2(E(X) − a)2
− exp −
.
(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
0
Z ∞
=E
1{(X−E(X))2 ≥s} ds
0
Z ∞
=
P (X − E(X))2 ≥ s ds
0
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(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
Eeλ(X−E(X))
b − E(X) λa E(X) − a λb
−λE(X)
≤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
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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, eL(λ) ,
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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
= a − E(X) + 00
L (λ) = II
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(E(X) − a)eλ(b−a)
0
L (λ) = a − E(X) + JJ
1−
E(X)−a
b−a
+
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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
(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
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Noting that
00
L (λ) ≤
(b − a)2
,
4
then by Taylor’s formula, we have
(b − a)2 2 (b − a)2 2
L(λ) ≤ L(0) + L (0)λ +
λ =
λ.
8
8
0
Therefore from Markov’s inequality, we have
√
√ (2.3)
P X − E(X) ≥ s ≤ inf e− sλ Eeλ(X−E(X))
λ>0
√
(b − a)2 2
≤ inf exp − sλ +
λ
λ>0
8
2s
= exp −
.
(b − a)2
Similarly,
P(−(X − E(X)) ≥
√
2s
s) ≤ exp −
(b − a)2
(E(X)−a)2
P −(X − E(X)) ≥
0
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0
+
vol. 8, iss. 3, art. 83, 2007
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E(X − E(X))2
Z (b−E(X))2
√ =
P (X − E(X)) ≥ s ds
Z
.
Yu Miao and Guangyu Yang
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From (2.2), it follows that
(2.4)
Upper Bounds for the
Dispersion
√ s ds
(b−E(X))2
Z (E(X)−a)2
2s
2s
≤
exp −
ds +
ds
exp −
(b − a)2
(b − a)2
0
0
(b − a)2
2(b − E(X))2
2(E(X) − a)2
=
− exp −
.
2 − exp −
2
(b − a)2
(b − a)2
Z
Furthermore,
2
2
2
2
2
2
E(X − E(X)) = E(X ) − (E(X)) ≤ E(X ) ≤ max{|a| , |b |},
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
which, by (2.4), implies the result.
vol. 8, iss. 3, art. 83, 2007
Remark 1. In fact, by,
s
2(E(X) − a)2
1
2(b − E(X))2
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(E(X) − a)2
2(b − E(X))2
M (a, b, E(X)) = √
− exp −
2 − exp −
(b − a)2
(b − a)2
2
p
1
= 1 − e−1/2 < √ ,
2
which means that under some assumptions, this bound is better than the result of [2]
for the case f ∈ L1 ([a, b]).
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Corollary 2.2. Let X be a random variable with a ≤ X ≤ b, then for any x ∈ [a, b],
(2.5)
p
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
2(E(X) − a)2
exp −
+ exp −
.
(b − a)2
(b − a)2
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
2
2
Proof. It is clear that (x − E(X)) ≤ (b − a) and from the proof of Theorem 2.1,
we know that
vol. 8, iss. 3, art. 83, 2007
σ 2 (X) + (x − E(X))2
1
2(E(X) − a)2
2(b − E(X))2
≤ 2−
exp −
+ exp −
(b − a)2 .
2
(b − a)2
(b − a)2
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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)}.
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3.
Lp Absolute Deviation
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,
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
(σp (X))p ≤ min{|b − a|p , M1 (a, b, p, X)},
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where
(b − a)2
2(|b − E(X)|p ∧ 1)
(3.1) 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
2(|a − E(X)|p ∧ 1)
(b − a)2
1 − exp −
+
2
(b − a)2
+ (|a − E(X)|p − |a − E(X)|p ∧ 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)},
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where
(3.2) M2 (a, b, p, X)
2|b − E(X)|p
(b − a)2
2|b − E(X)|p ∧ 1
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
Proof. On one hand, by the Fubini’s theorem, we have
Z |X−E(X)|p
p
(3.3)
E|X − E(X)| = E
ds
Z0 ∞
=E
1{|X−E(X)|p ≥s} ds
0
Z ∞
=
P(|X − E(X)|p ≥ s)ds
0
Z |b−E(X)|p
=
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
p
E|X − E(X)| =
P(X − E(X) ≥ s1/p )ds
0
Z |E(X)−a|p
+
P(−(X − E(X)) ≥ s1/p )ds
0
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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Z
≤
0
|b−E(X)|p
Z |E(X)−a|p
2s2/p
2s2/p
exp −
ds +
exp −
ds.
(b − a)2
(b − a)2
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
Z |b−E(X)|p ∧1
2s
2s2/p
≤
exp −
ds +
exp −
ds
(b − a)2
(b − a)2
|b−E(X)|p ∧1
0
(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 |E(X)−a|p
(b − a)2
2(|a − E(X)|p ∧ 1)
2s2/p
ds ≤
1 − exp −
exp −
(b − a)2
2
(b − a)2
0
2(|a − E(X)|p ∧ 1)2/p
p
p
+ (|a − E(X)| − |a − E(X)| ∧ 1) exp −
.
(b − a)2
(Case 1 ≤ p < 2) In this case, we have
Z |b−E(X)|p
2s2/p
ds
exp −
(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
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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(b − a)2
≤ |b − E(X)| ∧ 1 +
2
p
2|b − E(X)|p
2|b − E(X)|p ∧ 1
− exp −
exp −
(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
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
On the other hand, for any p ≥ 1, it is clear that
E|X − E(X)|p ≤ (b − a)p .
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From the above discussion, the desired results are obtained.
Remark 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
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References
[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].
Upper Bounds for the
Dispersion
Yu Miao and Guangyu Yang
vol. 8, iss. 3, art. 83, 2007
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