Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 188, 2006
A NOTE ON A WEIGHTED OSTROWSKI TYPE INEQUALITY FOR
CUMULATIVE DISTRIBUTION FUNCTIONS
A. RAFIQ AND N.S. BARNETT
C OMSATS I NSTITUTE OF I NFORMATION T ECHNOLOGY
I SLAMABAD
arafiq@comsats.edu.pk
S CHOOL OF C OMPUTER S CIENCE & M ATHEMATICS
V ICTORIA U NIVERSITY, PO B OX 14428
M ELBOURNE C ITY, MC 8001, AUSTRALIA .
neil@csm.vu.edu.au
Received 19 July, 2006; accepted 30 November, 2006
Communicated by S.S. Dragomir
A BSTRACT. In this note, a weighted Ostrowski type inequality for the cumulative distribution
function and expectation of a random variable is established.
Key words and phrases: Weight function, Ostrowski type inequality, Cumulative distribution functions.
2000 Mathematics Subject Classification. Primary 65D10, Secondary 65D15.
1. I NTRODUCTION
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,
Pr(X ≤ x) − 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
1 x − a+b
2
(1.1)
≤ +
2
(b − a)
for all x ∈ [a, b]. All the inequalities in (1.1) are sharp and the constant
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
192-06
1
2
the best possible.
2
A. R AFIQ AND N.S. BARNETT
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.
2. M AIN R ESULTS
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
Z
Taking ϕ (X) =
w(X)dX, then from (2.1) and integrating by parts, we get,
Z
w(X)dX
Z b Z
=
w(t)dt dF (t)
EW = E
a
Z
= W (b) −
(2.2)
b
w(t)F (t)dt,
a
where W (b) =
R
w(t)dt t=b .
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)
−
E
W
Pr(X ≤ x) −
m(a, b) Z b
1
≤
[m(a, x) − m(x, b)] Pr(X ≤ x) +
sgn(t − x)w(t)F (t)dt
m(a, b)
a
1
≤
[m(a, x) Pr(X ≤ x) + m(x, b) Pr(X ≥ x)]
m(a, b)
m(x,b)−m(a,x) 2
1 (2.3)
≤ +
2
m(a, b)
for all x ∈ [a, b]. All the inequalities in (2.3) are sharp and the constant
1
2
is the best possible.
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
J. Inequal. Pure and Appl. Math., 7(5) Art. 188, 2006
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI T YPE I NEQUALITY
3
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
Using (2.2) and (2.5), we get (see [2, p. 452]),
Z
m(a, b)F (x) + EW − W (b) =
(2.6)
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 monotonic
Rb
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 x Z t
Z b
Z b Z t
w(u)du dF (t) +
w(u)du dF (t)
p(x, t)dF (t) = a
a
x
b
a
Z x Z t
Z b Z t
≤
w(u)du dF (t) +
w(u)du dF (t)
a
a
x
b
x Z x
Z t
Z t
d
w(u)du F (t) −
F (t)
=
w(u)du dt
dt
a
a
a
a
b Z b
Z b
Z b
d
+
w(u)du F (t) −
F (t)
w(u)du dt
dt
t
x
t
x
Z b
(2.8)
= [m(a, x) − m(x, b)] F (x) +
sgn(t − x)w(t)F (t)dt.
a
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)
x
and
Z
b
sgn(t − x)w(t)F (t)dt ≤ m(x, b) for all x ∈ [a, b].
a
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.
J. Inequal. Pure and Appl. Math., 7(5) Art. 188, 2006
http://jipam.vu.edu.au/
4
A. R AFIQ AND N.S. BARNETT
Finally,
m(a, x) Pr(X ≤ x) + m(x, b) Pr(X ≥ x)
≤ 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 2.2. Since
Pr(X ≥ x) = 1 − Pr(X ≤ x),
we can obtain an equivalent to (2.3) for
E
+
m(a,
b)
−
W
(b)
W
.
Pr(X ≥ x) −
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 2.3.
a
+
b
W
(b)
−
E
W
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
w(t)F (t)dt
+
sgn t −
2
a
m( a+b
,b)−m(a, a+b
)
2
2
2
1 ≤ +
m(a, b)
2
and
a
+
b
E
+
m(a,
b)
−
W
(b)
W
Pr X ≥
−
2
m(a, b)
1
a+b
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
m( a+b
,b)−m(a, a+b
2
2 )
2
1 ≤ +
.
2
m(a, b)
Corollary 2.4.
"
#
a+b a+b
,
b
−
m
a,
2W
(b)
−
m(a,
b)
−
m
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
J. Inequal. Pure and Appl. Math., 7(5) Art. 188, 2006
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI T YPE I NEQUALITY
Additional inequalities for Pr [X ≤ x] and Pr X ≤
lary 2.6 and Remarks 2.5 and 2.7 of [1] using (2.3).
a+b
2
5
are obtainable in the style of Corol-
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 7(5) Art. 188, 2006
http://jipam.vu.edu.au/