volume 7, issue 5, article 186, 2006.
Received 22 April, 2006;
accepted 11 December, 2006.
Communicated by:C.E.M. Pearce
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Journal of Inequalities in Pure and Applied Mathematics
SOME P.D.F.-FREE UPPER BOUNDS FOR THE DISPERSION σ(X) AND THE QUANTITY σ2(X) + (x−EX)2
N. K. AGBEKO
Institute of Mathematics University of Miskolc
H-3515 Miskolc–Egyetemváros Hungary
EMail:matagbek@uni-miskolc.hu
c
2000Victoria University ISSN (electronic): 1443-5756 118-06
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
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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 dispersionσ(X) and the quantityσ2(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.
Contents
1 Introduction and Results. . . 3 References
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
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1. Introduction and Results
Let f : [a, b] ⊂ R → [0,∞) be the p.d.f. (probability density function) of a random variableX whose 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 dispersionσ(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,
1
p +1q = 1
√2 (b−a)
2 if f ∈L1[a, b]
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
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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 dispersion σ(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) that af(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).
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
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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 function h(t) = t2 decreases on (−∞, 0) and in- creases 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 that σ2(X) ≤ max{a2, b2}. Therefore, we can conclude on the validity of the argument.
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006
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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].
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.
Some p.d.f.-free Upper Bounds for the Dispersionσ(X)and the
Quantityσ2(X) + (x−EX)2 N. K. Agbeko
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References
[1] N.S.BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIO- TIS, 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].
