Journal of Inequalities in Pure and
Applied Mathematics
SOME P.D.F.-FREE UPPER BOUNDS FOR THE DISPERSION σ (X)
AND THE QUANTITY σ 2 (X) + (x − EX)2
volume 7, issue 5, article 186,
2006.
N. K. AGBEKO
Institute of Mathematics
University of Miskolc
H-3515 Miskolc–Egyetemváros
Hungary
Received 22 April, 2006;
accepted 11 December, 2006.
Communicated by: C.E.M. Pearce
EMail: matagbek@uni-miskolc.hu
Abstract
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Victoria University
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Abstract
In comparison with Theorems 2.1 and 2.4 in [1], we provide some p.d.f.-free
upper bounds for the dispersion σ (X) and the quantity σ2 (X) + (x − EX)2
taking only into account the endpoints of the given finite interval.
2000 Mathematics Subject Classification: 60E15, 26D15.
Key words: Dispersion, P.D.F.s.
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
Contents
1
Introduction and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
http://jipam.vu.edu.au
1.
Introduction and Results
Let f : [a, b] ⊂ R → [0, ∞) be the p.d.f. (probability density function) of a
random variable X whose expectation and dispersion are respectively given by
Z b
EX =
tf (t) dt
a
and
s
Z
σ (X) =
a
b
(t − EX)2 f (t) dt =
s
Z
b
t2 f (t) dt − (EX)2 .
a
In [1], Theorems 2.1 and 2.4, the following upper bounds were obtained for
the dispersion σ (X)
 √
3 (b − a)2


kf k∞
if f ∈ L∞ [a, b]



6




√
−1



2 (b − a)1+q

if f ∈ Lp [a, b] , p > 1,
2 kf kp
q
σ (X) ≤
2
[(q
+
1)
(2q
+
1)]

1


+ 1q = 1

p





√



 2 (b − a)
if f ∈ L1 [a, b]
2
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
http://jipam.vu.edu.au
and the quantity σ 2 (X) + (x − EX)2
σ 2 (X) + (x − EX)2

h
ip
(b−a)2
b+a 2


+
x
−
kf k∞ if f ∈ L∞ [a, b]
(b
−
a)

12
2





 h (b−x)2q+1 +(x−a)2q+1 i 2q1 q
kf kp
if f ∈ Lp [a, b] ,
2q+1
≤


p > 1, p1 + 1q = 1






 b−a + x − b+a 2
if f ∈ L1 [a, b]
2
2
for all x ∈ [a, b].
In this communication we intend to make free from the p.d.f. the above upper
bounds for the dispersion σ (X) and the quantity σ 2 (X) + (x − EX)2 taking
only into account the endpoints of the given finite interval.
Theorem 1.1. Under the above restriction on the p.d.f. we have
σ (X) ≤ min {max {|a| , |b|} , b − a} .
Proof. First, for any number t ∈ [a, b] we note (via f (t) ≥ 0) that af (t) ≤
tf (t) ≤ bf (t) leading to a ≤ EX ≤ b. Consequently,
(1.1)
0 ≤ EX − a ≤ b − a
and
0 ≤ b − EX ≤ b − a.
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
N. K. Agbeko
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We point out that the function g : [a, b] → [0, ∞), defined by g (t) = (t − EX)2 ,
is a bounded convex function which assumes the minimum at point (EX, 0).
J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
http://jipam.vu.edu.au
Thus
β := sup (t − EX)2 : t ∈ [a, b]
= max (a − EX)2 , (b − EX)2
≤ (b − a)2 ,
by taking into consideration (1.1). Now, it can be easily seen that
s
s
Z b
Z b
p
2
σ (X) =
(t − EX) f (t) dt ≤ β
f (t) dt = β ≤ b − a.
a
a
Next, using the facts that function h (t) = t2 decreases on (−∞, 0) and increases on (0, ∞) on the one hand and,
s
s
Z b
Z b
2
σ (X) =
t2 f (t) dt − (EX) ≤
t2 f (t) dt
a
a
on the other, we can easily check that
 2
b
if a ≥ 0


Z b

max {a2 , b2 } if a < 0 and b > 0
σ 2 (X) ≤
t2 f (t) dt ≤

a

 2
a
if b ≤ 0,
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
N. K. Agbeko
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so that σ 2 (X) ≤ max {a2 , b2 }. Therefore, we can conclude on the validity of
the argument.
J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
http://jipam.vu.edu.au
Theorem 1.2. Under the above restriction on the p.d.f. we have
q
σ 2 (X) + (x − EX)2 ≤ 2 min {max {|a| , |b|} , b − a}
for all x ∈ [a, b].
Proof. We recall the identity
2
2
Z
σ (X) + (x − EX) =
b
(t − x)2 f (t) dt,
x ∈ [a, b] ,
a
from the proof of Theorem 2.4 in [1]. Clearly,
Z b
(t − x)2 f (t) dt ≤ max (t − x)2 : t, x ∈ [a, b] ,
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
N. K. Agbeko
a
so that
q
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σ 2 (X) + (x − EX)2 ≤ max {|t − x| : t, x ∈ [a, b]} .
It is obvious that 0 ≤ t − a ≤ b − a and 0 ≤ x − a ≤ b − a, since t, x ∈ [a, b].
We note that we can estimate from above the quantity |t − x| in two ways:
|t − x| ≤ |t − a| + |a − x| ≤ 2 (b − a)
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and
|t − x| ≤ |t| + |x| ≤ 2 max {|a| , |b|} .
Consequently,
max {|t − x| : t, x ∈ [a, b]} ≤ 2 min {max {|a| , |b|} , b − a} .
This leads to the desired result.
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References
[1] 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].
Some p.d.f.-free Upper Bounds
for the Dispersion σ (X) and the
Quantity σ 2 (X) + (x − EX)2
N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
http://jipam.vu.edu.au