Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON A WEIGHTED OSTROWSKI TYPE INEQUALITY FOR
CUMULATIVE DISTRIBUTION FUNCTIONS
volume 7, issue 5, article 188,
2006.
A. RAFIQ AND N.S. BARNETT
Comsats Institute of Information Technology
Islamabad
EMail: arafiq@comsats.edu.pk
School of Computer Science & Mathematics
Victoria University, PO Box 14428
Melbourne City, MC 8001, Australia.
EMail: neil@csm.vu.edu.au
Received 19 July, 2006;
accepted 30 November, 2006.
Communicated by: S.S. Dragomir
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this note, a weighted Ostrowski type inequality for the cumulative distribution
function and expectation of a random variable is established.
2000 Mathematics Subject Classification: Primary 65D10, Secondary 65D15.
Key words: Weight function, Ostrowski type inequality, Cumulative distribution functions.
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
A. Rafiq and N.S. Barnett
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J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006
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1.
Introduction
In [1], N. S. Barnett and S. S. Dragomir established the following Ostrowski
type inequality for cumulative distribution functions.
Theorem 1.1. Let X be a random variable taking values in the finite interval
[a, b], with cumulative distribution function F (x) = Pr(X ≤ x), then,
b − E(X)
b−a
Z b
1
[2x − (a + b)] Pr(X ≤ x) +
sgn(t − x)F (t)dt
≤
b−a
a
1
≤
[(b − x)) Pr(X ≥ x) + (x − a) Pr(X ≤ x)]
b−a
x − a+b
1
2
≤ +
(b − a)
2
Pr(X ≤ x) −
(1.1)
for all x ∈ [a, b]. All the inequalities in (1.1) are sharp and the constant
best possible.
1
2
the
In this paper, we establish a weighted version of this result using similar
methods to those used in [1]. The results of [1] are then retrieved by taking the
weight function to be 1.
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
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2.
Main Results
We assume that the weight function w : (a, b) −→ [0, ∞), is integrable, nonnegative and
Z b
w(t)dt < ∞.
a
The domain of w may be finite or infinite and w may vanish at the boundary
points. We denote
Z b
m(a, b) =
w(t)dt.
a
We also know that the expectation of any function ϕ(X) of the random variable
X is given by:
Z b
(2.1)
E [ϕ (X)] =
ϕ(t)dF (t) .
a
w(X)dX, then from (2.1) and integrating by parts, we get,
Z
EW = E
w(X)dX
Z b Z
=
w(t)dt dF (t)
a
Z b
= W (b) −
w(t)F (t)dt,
(2.2)
A. Rafiq and N.S. Barnett
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Z
Taking ϕ (X) =
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
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a
where W (b) =
R
w(t)dt t=b .
J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006
http://jipam.vu.edu.au
Theorem 2.1. Let X be a random variable taking values in the finite interval
[a, b], with cumulative distribution function F (x) = P r(X ≤ x), then,
W (b) − EW
m(a, b)
1
≤
[m(a, x) − m(x, b)] Pr(X ≤ x)
m(a, b)
Z b
+
sgn(t − x)w(t)F (t)dt
Pr(X ≤ x) −
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
a
≤
(2.3)
≤
1
[m(a, x) Pr(X ≤ x) + m(x, b) Pr(X ≥ x)]
m(a, b)
1
+
2
A. Rafiq and N.S. Barnett
m(x,b)−m(a,x)
2
m(a, b)
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for all x ∈ [a, b]. All the inequalities in (2.3) are sharp and the constant
best possible.
1
2
is the
Proof. Consider the Kernel p : [a, b]2 −→ R defined by
 Rt
 a w(u)du if t ∈ [a, x]
(2.4)
p(x, t) =
 Rt
w(u)du if t ∈ (x, b],
b
Rb
then the Riemann-Stieltjes integral a p(x, t)dF (t) exists for any x ∈ [a, b] and
the following identity holds:
Z b
Z b
(2.5)
p(x, t)dF (t) = m(a, b)F (x) −
w(t)F (t)dt .
a
a
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Using (2.2) and (2.5), we get (see [2, p. 452]),
Z
(2.6)
m(a, b)F (x) + EW − W (b) =
b
p(x, t)dF (t) .
a
As shown in [1], if p : [a, b] → R is continuous on [a, b] and ν : [a, b] → R is
Rb
monotonic non-decreasing, then the Riemann -Stieltjes integral a p (x) dν (x)
exists and
Z b
Z b
(2.7)
p (x) dν (x) ≤
|p (x)| dν (x) .
a
a
Using (2.7) we have
Z b
p(x, t)dF (t)
a
Z x Z t
Z b Z t
=
w(u)du dF (t) +
w(u)du dF (t)
a
a
x
b
Z x Z t
Z b Z t
≤
w(u)du dF (t) +
w(u)du dF (t)
a
a
x
b
Z t
Z t
Z x
x
d
=
w(u)du F (t) −
F (t)
w(u)du dt
dt
a
a
a
a
Z b
Z b
Z b
b
d
w(u)du dt
+
w(u)du F (t) −
F (t)
dt
t
t
x
x
Z b
(2.8)
= [m(a, x) − m(x, b)] F (x) +
sgn(t − x)w(t)F (t)dt.
a
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
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Using the identity (2.6) and the inequality (2.8), we deduce the first part of (2.3).
We know that
Z b
Z x
Z b
sgn(t − x)w(t)F (t)dt = −
w(t)F (t)dt +
w(t)F (t)dt.
a
a
x
As F (·) is monotonic non-decreasing on [a, b],
Z x
w(t)F (t)dt ≥ m(a, x)F (a) = 0,
a
Z
b
w(t)F (t)dt ≤ m(x, b)F (b) = m(x, b)
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
x
and
Z
b
sgn(t − x)w(t)F (t)dt ≤ m(x, b) for all x ∈ [a, b].
a
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Consequently,
Z
[m(a, x) − m(x, b)] Pr (X ≤ x) +
b
sgn(t − x)w(t)F (t)dt
a
≤ [m(a, x) − m(x, b)] Pr(X ≤ x) + m(x, b)
= m(a, x) Pr(X ≤ x) + m(x, b) Pr(X ≥ x)
and the second part of (2.3) is proved.
Finally,
m(a, x) Pr(X ≤ x) + m(x, b) Pr(X ≥ x)
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≤ max {m(a, x), m(x, b)} [Pr(X ≤ x) + Pr(X ≥ x)]
m(a, b) + |m(x, b) − m(a, x)|
=
2
and the last part of (2.3) follows.
Remark 1. Since
Pr(X ≥ x) = 1 − Pr(X ≤ x),
we can obtain an equivalent to (2.3) for
Pr(X ≥ x) −
EW + m(a, b) − W (b)
.
m(a, b)
Following the same style of argument as in Remark 2.3 and Corollary 2.4 of
[1], we have the following two corollaries.
Corollary 2.2.
W (b) − EW
a+b
Pr X ≤
−
2
m(a, b)
1
a+b
a+b
≤
m a,
−m
,b
Pr(X ≤ x)
m(a, b)
2
2
Z b
a+b
+
sgn t −
w(t)F (t)dt
2
a
m( a+b
,b)−m(a, a+b
2
2 )
2
1
≤ +
2
m(a, b)
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
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and
EW + m(a, b) − W (b)
a+b
−
Pr X ≥
2
m(a, b)
a+b
1
a+b
a+b
≤
m a,
−m
,b
Pr X ≤
m(a, b)
2
2
2
Z b
a+b
+
sgn t −
w(t)F (t)dt
2
a
,b)−m(a, a+b
m( a+b
2
2 )
2
1
.
≤ +
m(a, b)
2
Corollary 2.3.
"
#
2W (b) − m(a, b) − m a+b
, b − m a, a+b
1
2
2
− EW
m(a, b)
2
a+b
≤ Pr X ≤
2
"
#
a+b
2W (b) − m(a, b) + m a+b
,
b
−
m
a,
1
2
2
≤1+
− EW .
m(a, b)
2
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
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Additional inequalities for Pr [X ≤ x] and Pr X ≤ a+b
are obtainable in
2
the style of Corollary 2.6 and Remarks 2.5 and 2.7 of [1] using (2.3).
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References
[1] N.S. BARNETT AND S.S. DRAGOMIR, An inequality of Ostrowski’s type
for cumulative distribution functions, Kyungpook Math. J., 39(2) (1999),
303–311.
[2] S.S. DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, 2002.
A Note on a Weighted Ostrowski
Type Inequality For Cumulative
Distribution Functions
A. Rafiq and N.S. Barnett
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