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 σ2(X) + (x−EX)2
N. K. AGBEKO INSTITUTE OFMATHEMATICS
UNIVERSITY OFMISKOLC
H-3515 MISKOLC–EGYETEMVÁROS
HUNGARY
Received 22 April, 2006; accepted 11 December, 2006 Communicated by C.E.M. Pearce
ABSTRACT. In comparison with Theorems 2.1 and 2.4 in [1], we provide some p.d.f.-free upper bounds for the dispersionσ(X)and the quantityσ2(X) + (x−EX)2taking 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. INTRODUCTION ANDRESULTS
Let f : [a, b]⊂R→[0,∞)be the p.d.f. (probability density function) of a random variable Xwhose expectation and dispersion are respectively given by
EX = Z b
a
tf(t)dt and
σ(X) = s
Z b a
(t−EX)2f(t)dt= s
Z b a
t2f(t)dt−(EX)2.
In [1], Theorems 2.1 and 2.4, the following upper bounds were obtained for the dispersion σ(X)
σ(X)≤
√3(b−a)2
6 kfk∞ if f ∈L∞[a, b]
√2(b−a)1+q−1 2[(q+1)(2q+1)]2q
kfkp if f ∈Lp[a, b], p >1,1p +1q = 1
√2(b−a)
2 if f ∈L1[a, b]
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
118-06
2 N. K. AGBEKO
and the quantityσ2(X) + (x−EX)2 σ2(X) + (x−EX)2
≤
(b−a)h(b−a)2
12 + x− b+a2 2i
pkfk∞ if f ∈L∞[a, b]
h(b−x)2q+1+(x−a)2q+1
2q+1
i2q1 q
kfkp if f ∈Lp[a, b], p >1, 1p + 1q = 1
b−a
2 +
x− b+a2
2
if f ∈L1[a, b]
for allx∈[a, b].
In this communication we intend to make free from the p.d.f. the above upper bounds for the dispersionσ(X)and the quantityσ2(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) thataf(t) ≤ tf(t) ≤ bf(t) leading toa≤EX ≤b. Consequently,
(1.1) 0≤EX−a≤b−a and 0≤b−EX ≤b−a.
We point out that the functiong : [a, b]→[0,∞), defined byg(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
σ(X) = s
Z b a
(t−EX)2f(t)dt ≤ s
β Z b
a
f(t)dt =p
β ≤b−a.
Next, using the facts that functionh(t) = t2 decreases on (−∞, 0)and increases on (0, ∞) on the one hand and,
σ(X) = s
Z b a
t2f(t)dt−(EX)2 ≤ s
Z b a
t2f(t)dt
on the other, we can easily check that
σ2(X)≤ Z b
a
t2f(t)dt≤
b2 if a≥0
max{a2, b2} if a <0andb > 0
a2 if b≤0,
so thatσ2(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
σ2(X) + (x−EX)2 ≤2 min{max{|a|,|b|}, b−a}
for allx∈[a, b].
J. Inequal. Pure and Appl. Math., 7(5) Art. 186, 2006 http://jipam.vu.edu.au/
SOME P.D.F.-FREE UPPER BOUNDS 3
Proof. We recall the identity
σ2(X) + (x−EX)2 = Z b
a
(t−x)2f(t)dt, x∈[a, b], from the proof of Theorem 2.4 in [1]. Clearly,
Z b a
(t−x)2f(t)dt≤max
(t−x)2 :t, x∈[a, b] , so that
q
σ2(X) + (x−EX)2 ≤max{|t−x|:t, x∈[a, b]}.
It is obvious that0≤t−a ≤b−aand0≤ x−a ≤ b−a, sincet, 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.
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].
J. Inequal. Pure and Appl. Math., 7(5) Art. 186, 2006 http://jipam.vu.edu.au/
