JJ II

volume 7, issue 5, article 188, 2006.

Received 19 July, 2006;

accepted 30 November, 2006.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON A WEIGHTED OSTROWSKI TYPE INEQUALITY FOR CUMULATIVE DISTRIBUTION FUNCTIONS

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

2000c Victoria University ISSN (electronic): 1443-5756 192-06

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

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 func- tions.

Contents

1 Introduction. . . 3 2 Main Results . . . 4

References

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page3of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

1. Introduction

In [1], N. S. Barnett and S. S. Dragomir established the following Ostrowski type inequality for cumulative distribution functions.

Theorem 1.1. LetX be a random variable taking values in the finite interval [a, b], with cumulative distribution functionF(x) = Pr(X ≤x), then,

Pr(X ≤x)− b−E(X) b−a

≤ 1 b−a

[2x−(a+b)] Pr(X≤x) + Z b

a

sgn(t−x)F(t)dt

≤ 1

b−a[(b−x)) Pr(X ≥x) + (x−a) Pr(X ≤x)]

≤ 1 2 +

x− ^a+b₂ (b−a) (1.1)

for all x ∈ [a, b].All the inequalities in (1.1) are sharp and the constant ¹₂ the best possible.

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 be1.

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

2. Main Results

We assume that the weight function w : (a, b) −→ [0,∞),is integrable, non- negative and

Z b a

w(t)dt <∞.

The domain of w may be finite or infinite andw may vanish at the boundary points. We denote

m(a, b) = Z b

a

w(t)dt.

We also know that the expectation of any functionϕ(X)of the random variable X is given by:

(2.1) E[ϕ(X)] =

Z b a

ϕ(t)dF(t). Takingϕ(X) =

Z

w(X)dX,then from (2.1) and integrating by parts, we get, E_W =E

Z

w(X)dX

= Z b

a

Z

w(t)dt

dF (t)

=W(b)− Z b

a

w(t)F(t)dt, (2.2)

whereW(b) =R

w(t)dt

t=b.

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

Theorem 2.1. LetX be a random variable taking values in the finite interval [a, b], with cumulative distribution functionF(x) =P r(X ≤x),then,

Pr(X ≤x)−W(b)−E_W m(a, b)

≤ 1 m(a, b)

[m(a, x)−m(x, b)] Pr(X ≤x) +

Z b a

sgn(t−x)w(t)F(t)dt

≤ 1

m(a, b)[m(a, x) Pr(X ≤x) +m(x, b) Pr(X ≥x)]

≤ 1 2+

m(x,b)−m(a,x) 2

m(a, b) (2.3)

for allx∈[a, b].All the inequalities in (2.3) are sharp and the constant ¹₂ is the best possible.

Proof. Consider the Kernelp: [a, b]² −→R defined by

(2.4) p(x, t) =





 Rt

aw(u)du ift ∈[a, x]

Rt

b w(u)du ift ∈(x, b], then the Riemann-Stieltjes integralRb

ap(x, t)dF (t)exists for anyx∈[a, b]and the following identity holds:

(2.5)

Z b a

p(x, t)dF (t) = m(a, b)F(x)− Z b

a

w(t)F(t)dt.

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

Using (2.2) and (2.5), we get (see [2, p. 452]), (2.6) m(a, b)F(x) +E_W −W(b) =

Z b a

p(x, t)dF (t).

As shown in [1], ifp: [a, b]→Ris continuous on[a, b]andν: [a, b]→Ris monotonic non-decreasing, then the Riemann -Stieltjes integralRb

a p(x) dν(x) exists and

(2.7)

Z b a

p(x) dν(x)

≤ Z b

a

|p(x)|dν(x). Using (2.7) we have

Z b a

p(x, t)dF (t)

=

Z x a

Z t a

w(u)du

dF (t) + Z b

x

Z t b

w(u)du

dF (t)

≤ Z x

a

Z t a

w(u)du

dF (t) + Z b

x

Z t b

w(u)du

dF (t)

= Z t

a

w(u)du

F(t)

x

a

− Z x

a

F(t)d dt

Z t a

w(u)du

dt

+ Z b

t

w(u)du

F(t)

b

x

− Z b

x

F(t)d dt

Z b t

w(u)du

dt

= [m(a, x)−m(x, b)]F(x) + Z b

a

sgn(t−x)w(t)F(t)dt.

(2.8)

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

Using the identity (2.6) and the inequality (2.8), we deduce the first part of (2.3).

We know that Z b

a

sgn(t−x)w(t)F(t)dt=− Z x

a

w(t)F(t)dt+ Z b

x

w(t)F(t)dt.

AsF(·)is monotonic non-decreasing on[a, b], Z x

a

w(t)F(t)dt≥m(a, x)F(a) = 0, Z b

x

w(t)F(t)dt≤m(x, b)F(b) = m(x, b) and

Z b a

sgn(t−x)w(t)F(t)dt≤m(x, b)for allx∈[a, b].

Consequently,

[m(a, x)−m(x, b)] Pr (X ≤x) + Z b

a

sgn(t−x)w(t)F(t)dt

≤[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)

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

≤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)− E_W +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.

Pr

X ≤ a+b 2

−W(b)−EW

m(a, b)

≤ 1 m(a, b)

m

a,a+b 2

−m

a+b 2 , b

Pr(X ≤x) +

Z b a

sgn

t−a+b 2

w(t)F(t)dt

≤ 1 2 +

m(^a+b2 ,b)^−m(^a,a+b2 )

2

m(a, b)

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

and

Pr

X ≥ a+b 2

− E_W +m(a, b)−W(b) m(a, b)

≤ 1 m(a, b)

m

a,a+b 2

−m

a+b 2 , b

Pr

X ≤ a+b 2

+ Z b

a

sgn

t− a+b 2

w(t)F(t)dt

≤ 1 2 +

m(^a+b2 ,b)^−m(^a,a+b2 )

2

m(a, b) . Corollary 2.3.

1 m(a, b)

"

2W(b)−m(a, b)−

m ^a+b₂ , b

−m a,^a+b₂

2 −EW

#

≤Pr

X ≤ a+b 2

≤1 + 1 m(a, b)

"

2W(b)−m(a, b) +

m ^a+b₂ , b

−m a,^a+b₂

2 −E_W

# .

Additional inequalities forPr [X ≤x]andPr X ≤ ^a+b₂

are obtainable in the style of Corollary 2.6 and Remarks 2.5 and 2.7 of [1] using (2.3).

A Note on a Weighted Ostrowski Type Inequality For Cumulative

Distribution Functions A. Rafiq and N.S. Barnett

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of10

J. Ineq. Pure and Appl. Math. 7(5) Art. 188, 2006

http://jipam.vu.edu.au

References

[1] N.S. BARNETTANDS.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 Inequali- ties and Applications in Numerical Integration, Kluwer Academic Publish- ers, 2002.