Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 186, 2006
SOME P.D.F.-FREE UPPER BOUNDS FOR THE DISPERSION σ (X) AND THE
QUANTITY σ 2 (X) + (x − EX)2
N. K. AGBEKO
I NSTITUTE OF M ATHEMATICS
U NIVERSITY OF M ISKOLC
H-3515 M ISKOLC –E GYETEMVÁROS
H UNGARY
matagbek@uni-miskolc.hu
Received 22 April, 2006; accepted 11 December, 2006
Communicated by C.E.M. Pearce
A BSTRACT. In comparison with Theorems 2.1 and 2.4 in [1], we provide some p.d.f.-free upper
2
bounds for the dispersion σ (X) and the quantity σ 2 (X) + (x − EX) taking only into account
the endpoints of the given finite interval.
Key words and phrases: Dispersion, P.D.F.s.
2000 Mathematics Subject Classification. 60E15, 26D15.
1. I NTRODUCTION AND R ESULTS
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
s
b
2
Z
(t − EX) f (t) dt =
σ (X) =
a
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, p1 + 1q = 1
σ (X) ≤
2 kf kp
q

2[(q+1)(2q+1)]

 √

 2(b−a)
if f ∈ L1 [a, b]
2
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
118-06
2
N. K. AGBEKO
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
i1 q
(b−x)2q+1 +(x−a)2q+1 2q
≤
kf kp
if f ∈ Lp [a, b] , p > 1,
2q+1





 b−a b+a 2
+
x
−
if f ∈ L1 [a, b]
2
2
1
p
+
1
q
=1
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,
0 ≤ EX − a ≤ b − a
(1.1)
and
0 ≤ b − EX ≤ b − a.
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). 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
2
σ (X) =
t f (t) dt − (EX) ≤
t2 f (t) dt
a
a
on the other, we can easily check that
 2
b
if a ≥ 0



b
max {a2 , b2 } if a < 0 and b > 0
σ 2 (X) ≤
t2 f (t) dt ≤

a

 2
a
if b ≤ 0,
Z
so that σ 2 (X) ≤ max {a2 , b2 }. Therefore, we can conclude on the validity of the argument.
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].
J. Inequal. Pure and Appl. Math., 7(5) Art. 186, 2006
http://jipam.vu.edu.au/
S OME P. D . F.- FREE UPPER BOUNDS
3
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] ,
a
so that
q
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)
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.
R EFERENCES
[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].
J. Inequal. Pure and Appl. Math., 7(5) Art. 186, 2006
http://jipam.vu.edu.au/