Journl of Inequlities in Pure nd Applied Mthemtics SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARI- ABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR AND JOHN ROUMELIOTIS School of Communictions nd Informtics Victori University of Technology PO Box 1448, Melbourne City MC 8001 Victori, Austrli EMil: neil@mtild.vu.edu.u EMil: pc@mtild.vu.edu.u EMil: sever@mtild.vu.edu.u EMil: johnr@mtild.vu.edu.u volume, issue 1, rticle 1, 001. Received 7 Jnury, 000; ccepted 16 June 000. Communicted by: C.E.M. Perce Abstrct Home Pge Go Bck c 000 School of Communictions nd Informtics,Victori University of Technology ISSN (electronic): 1443-5756 00-99
Abstrct Some inequlities for the dispersion of rndom vrible whose pdf is defined on finite intervl nd pplictions re given. 000 Mthemtics Subject Clssifiction: 60E15, 6D15 Key words: Rndom vrible, Expecttion, Vrince, Dispersion, Grüss Inequlity, Chebychev s Inequlity, Lupş Inequlity. 1 Introduction......................................... 3 Some Inequlities for Dispersion........................ 4 3 Perturbed Results Using Grüss Type inequlities.......... 1 3.1 Perturbed Results Using Premture Inequlities... 13 3. Alternte Grüss Type Results for Inequlities Involving the Vrince........................... 18 4 Some Inequlities for Absolutely Continuous P.D.F s...... 4 References Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
1. Introduction In this note we obtin some inequlities for the dispersion of continuous rndom vrible X hving the probbility density function (p.d.f.) f defined on finite intervl [, b]. Tools used include: Korkine s identity, which plys centrl role in the proof of Chebychev s integrl inequlity for synchronous mppings [4], Hölder s weighted inequlity for double integrls nd n integrl identity connecting the vrince σ (X) nd the expecttion E (X). Perturbed results re lso obtined by using Grüss, Chebyshev nd Lupş inequlities. In Section 4, results from n identity involving double integrl re obtined for vriety of norms. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
. Some Inequlities for Dispersion Let f : [, b] R R + be the p.d.f. of the rndom vrible X nd E (X) : its expecttion nd [ ] 1 σ (X) (t E (X)) f (t) dt its dispersion or stndrd devition. The following theorem holds. tf (t) dt [ Theorem.1. With the bove ssumptions, we hve (.1) 0 σ (X) 3(b ) t f (t) dt [E (X)] ] 1 6 f, provided f L, [, b] ; (b ) 1+ 1 q [(q+1)(q+1)] q (b ). Proof. Korkine s identity [4], is (.) p (t) dt 1 p (t) g (t) h (t) dt f p, provided f L p [, b] nd p > 1, 1 p + 1 q 1; p (t) g (t) dt p (t) h (t) dt p (t) p (s) (g (t) g (s)) (h (t) h (s)) dtds, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 4 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
which holds for the mesurble mppings p, g, h : [, b] R for which the integrls involved in (.) exist nd re finite. Choose in (.) p (t) f (t), g (t) h (t) t E (X), t [, b] to get (.3) σ (X) 1 It is obvious tht (.4) f (t) f (s) (t s) dtds f (t) f (s) (t s) dtds. sup f (t) f (s) (t,s) [,b] (t s) dtds (b )4 f 6 nd then, by (.3), we obtin the first prt of (.1). For the second prt, we pply Hölder s integrl inequlity for double integrls to obtin f (t) f (s) (t s) dtds ( f p ) 1 ( f p (t) f p p b (s) dtds [ ] (b ) q+ 1 q, (q + 1) (q + 1) ) 1 (t s) q q dtds Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 5 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
where p > 1 nd 1 + 1 1, nd the second inequlity in (.1) is proved. p q For the lst prt, observe tht f (t) f (s) (t s) dtds sup (t,s) [,b] (t s) (b ) f (t) f (s) dtds s f (t) f (s) dtds f (t) dt f (s) ds 1. Using finer rgument, the lst inequlity in (.1) cn be improved s follows. Theorem.. Under the bove ssumptions, we hve (.5) 0 σ (X) 1 (b ). Proof. We use the following Grüss type inequlity: (.6) 0 p (t) ( ) g (t) dt p (t) g (t) dt 1 p (t) dt p (t) dt 4 (M m), provided tht p, g re mesurble on [, b] nd ll the integrls in (.6) exist nd re finite, p (t) dt > 0 nd m g M.e. on [, b]. For proof of this inequlity see [19]. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 6 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Choose in (.6), p (t) f (t), g (t) t E (X), t [, b]. Observe tht in this cse m E (X), M b E (X) nd then, by (.6) we deduce (.5). Remrk.1. The sme conclusion cn be obtined for the choice p (t) f (t) nd g (t) t, t [, b]. The following result holds. Theorem.3. Let X be rndom vrible hving the p.d.f. given by f : [, b] R R +. Then for ny x [, b] we hve the inequlity: (.7) σ (X) + (x E (X)) [ (b ) (b ) + ( ) ] x +b f 1, provided f L [, b] ; [ (b x) q+1 +(x ) q+1 ( b q+1 + x +b Proof. We observe tht (.8) ). (x t) f (t) dt ] 1 q f p, provided f L p [, b], p > 1, nd 1 p + 1 q 1; ( x xt + t ) f (t) dt x xe (X) + t f (t) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 7 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
nd s (.9) σ (X) we get, by (.8) nd (.9), t f (t) dt [E (X)], (.10) [x E (X)] + σ (X) which is of interest in itself too. We observe tht (x t) f (t) dt ess sup f (t) t [,b] (x t) f (t) dt, (x t) dt f (b x) 3 + (x ) 3 (b ) f [ 3 (b ) 1 + ( x + b ) ] nd the first inequlity in (.7) is proved. For the second inequlity, observe tht by Hölder s integrl inequlity, ( ) 1 ( (x t) f (t) dt f p p b ) 1 (t) dt (x t) q q dt [ ] (b x) q+1 + (x ) q+1 1 q f p, q + 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 8 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
nd the second inequlity in (.7) is estblished. Finlly, observe tht, nd the theorem is proved. (x t) f (t) dt sup (x t) f (t) dt t [,b] mx { (x ), (b x) } The following corollries re esily deduced. (mx {x, b x}) ( b + x + b ), Corollry.4. With the bove ssumptions, we hve (.11) 0 σ (X) (b ) 1 [ (b ) + ( E (X) +b 1 [ (b E(X)) q+1 +(E(X) ) q+1 q+1 ) ] 1 f 1, provided f L [, b] ; ] 1 q f 1 p, if f L p [, b], p > 1 nd 1 p + 1 q 1; + E (X) +b. b Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 9 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Remrk.. The lst inequlity in (.1) is worse thn the inequlity (.5), obtined by technique bsed on Grüss inequlity. The best inequlity we cn get from (.7) is tht one for which x +b, nd this pplies for ll the bounds since nd min x [,b] [ (b ) 1 + ( x + b ) ] (b x) q+1 + (x ) q+1 min x [,b] q + 1 [ b min + x [,b] x + b ] (b ) 1 (b )q+1 q (q + 1), b. Consequently, we cn stte the following corollry s well. Corollry.5. With the bove ssumptions, we hve the inequlity: (.1) 0 σ (X) + [ E (X) + b ] (b ) 3 1 f, provided f L [, b] ; (b ) q+1 4(q+1) 1 q (b ) 4. f p, if f L p [, b], p > 1, nd 1 p + 1 q 1;, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 10 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Remrk.3. From the lst inequlity in (.1), we obtin (.13) 0 σ (X) (b E (X)) (E (X) ) 1 4 (b ), which is n improvement on (.5). Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 11 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
3. Perturbed Results Using Grüss Type inequlities In 1935, G. Grüss (see for exmple [6]) proved the following integrl inequlity which gives n pproximtion for the integrl of product in terms of the product of the integrls. Theorem 3.1. Let h, g : [, b] R be two integrble mppings such tht φ h (x) Φ nd γ g (x) Γ for ll x [, b], where φ, Φ, γ, Γ re rel numbers. Then, (3.1) T (h, g) 1 (Φ φ) (Γ γ), 4 where (3.) T (h, g) 1 b 1 b h (x) g (x) dx h (x) dx 1 b g (x) dx nd the inequlity is shrp, in the sense tht the constnt 1 cnnot be replced 4 by smller one. For simple proof of this s well s for extensions, generlistions, discrete vrints nd other ssocited mteril, see [5], nd [1]-[1] where further references re given. A premture Grüss inequlity is embodied in the following theorem which ws proved in [3]. It provides shrper bound thn the bove Grüss inequlity. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 1 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Theorem 3.. Let h, g be integrble functions defined on [, b] nd let d g (t) D. Then (3.3) T (h, g) D d where T (h, g) is s defined in (3.). T (h, h) 1, Theorem 3. will now be used to provide perturbed rule involving the vrince nd men of p.d.f. 3.1. Perturbed Results Using Premture Inequlities In this subsection we develop some perturbed results. Theorem 3.3. Let X be rndom vrible hving the p.d.f. given by f : [, b] R R +. Then for ny x [, b] nd m f (x) M we hve the inequlity (3.4) P V (x) : σ (X) + (x E (X)) M m (M m) (b ) 45 (b )3 45. [ (b (b ) 1 ( x + b ) ) ( + 15 x + b ) ] 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 13 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Proof. Applying the premture Grüss result (3.3) by ssociting g (t) with f (t) nd h (t) (x t), gives, from (3.1)-(3.3) (3.5) where from (3.) (x t) f (t) dt 1 b (3.6) T (h, h) 1 b Now, (3.7) nd 1 b 1 b (x t) 4 dt (x t) dt f (t) dt (b ) M m [ 1 b [T (h, h)] 1, (x t) dt]. (x t) dt (x )3 + (b x) 3 3 (b ) 1 ( ) ( b + x + b 3 (x t) 4 dt (x )5 + (b x) 5 5 (b ) giving, for (3.6), [ ] [ ] (x ) 5 + (b x) 5 (x ) 3 + (b x) 3 (3.8) 45T (h, h) 9 5. b b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 14 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Let A x nd B b x in (3.8) to give ( ) ( A 5 + B 5 A 3 + B 3 45T (h, h) 9 5 A + B A + B 9 [ A 4 A 3 B + A B AB 3 + B 4] 5 [ A AB + B ] ) ( 4A 7AB + 4B ) (A + B) [ (A ) ( ) ] + B A B + 15 (A + B). Using the fcts tht A + B b nd A B x ( + b) gives [ (b ) ( (b ) (3.9) T (h, h) + 15 x + b ) ] 45 nd from (3.7) giving (3.10) 1 b 1 b (x t) dt A3 + B 3 3 (A + B) 1 [ A AB + B ] 3 [ (A 1 ) ( ) ] + B A B + 3, 3 (x t) dt ( (b ) + x + b ). 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 15 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Hence, from (3.5), (3.9) (3.10) nd (.10), the first inequlity in (3.4) results. The corsest uniform bound is obtined by tking x t either end point. Thus the theorem is completely proved. Remrk 3.1. The best inequlity obtinble from (3.4) is t x +b giving [ (3.11) σ (X) + E (X) + b ] (b ) 1 M m (b ) 3. 1 5 The result (3.11) is tighter bound thn tht obtined in the first inequlity of (.1) since 0 < M m < f. For symmetric p.d.f. E (X) +b nd so the bove results would give bounds on the vrince. The following results hold if the p.d.f f (x) is differentible, tht is, for f (x) bsolutely continuous. Theorem 3.4. Let the conditions on Theorem 3.1 be stisfied. Further, suppose tht f is differentible nd is such tht Then f : sup f (t) <. t [,b] (3.1) P V (x) b 1 f I (x), where P V (x) is given by the left hnd side of (3.4) nd, [ (b ) ( (b ) (3.13) I (x) + 15 x + b ) ] 1. 45 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 16 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Proof. Let h, g : [, b] R be bsolutely continuous nd h, g be bounded. Then Chebychev s inequlity holds (see [3]) T (h, g) (b ) 1 sup h (t) sup g (t). t [,b] t [,b] Mtić, Pečrić nd Ujević [3] using premture Grüss type rgument proved tht (3.14) T (h, g) (b ) 1 sup g (t) T (h, h). t [,b] Associting f ( ) with g ( ) nd (x ) with h ( ) in (3.13) gives, from (3.5) nd (3.9), I (x) (b ) [T (h, h)] 1, which simplifies to (3.13) nd the theorem is proved. Theorem 3.5. Let the conditions of Theorem 3.3 be stisfied. Further, suppose tht f is loclly bsolutely continuous on (, b) nd let f L (, b). Then (3.15) P V (x) b π f I (x), where P V (x) is the left hnd side of (3.4) nd I (x) is s given in (3.13). Proof. The following result ws obtined by Lupş (see [3]). For h, g : (, b) R loclly bsolutely continuous on (, b) nd h, g L (, b), then T (h, g) (b ) π h g, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 17 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
where k : ( 1 b k (t) ) 1 Mtić, Pečrić nd Ujević [3] further show tht for k L (, b). (3.16) T (h, g) b π g T (h, h). Associting f ( ) with g ( ) nd (x ) with h in (3.16) gives (3.15), where I (x) is s found in (3.13), since from (3.5) nd (3.9), I (x) (b ) [T (h, h)] 1. 3.. Alternte Grüss Type Results for Inequlities Involving the Vrince Let (3.17) S (h (x)) h (x) M (h) where (3.18) M (h) 1 b Then from (3.), h (u) du. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 18 of 41 (3.19) T (h, g) M (hg) M (h) M (g). J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Drgomir nd McAndrew [19] hve shown, tht (3.0) T (h, g) T (S (h), S (g)) nd proceeded to obtin bounds for trpezoidl rule. Identity (3.0) is now pplied to obtin bounds for the vrince. Theorem 3.6. Let X be rndom vrible hving the p.d.f. f : [, b] R R +. Then for ny x [, b] the following inequlity holds, nmely, (3.1) P V (x) 8 3 ν3 (x) f ( ) 1 b if f L [, b], where P V (x) is s defined by the left hnd side of (3.4), nd ν ν (x) ( 1 b ) ( ) 3 + x +b. Proof. Using identity (3.0), ssocite with h ( ), (x ) nd f ( ) with g ( ). Then (3.) where from (3.18), (x t) f (t) dt M ( (x ) ) M ( (x ) ) 1 b [ (x t) M ( (x ) )] [ f (t) 1 ] dt, b (x t) dt 1 [ (x ) 3 + (b x) 3] 3 (b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 19 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
nd so (3.3) 3M ( (x ) ) ( ) ( b + 3 x + b ). Further, from (3.17), nd so, on using (3.3) S ( (x ) ) (x t) M ( (x ) ) (3.4) S ( (x ) ) (x t) 1 3 ( b ) ( x + b ). Now, from (3.) nd using (.10), (3.3) nd (3.4), the following identity is obtined [ (b (3.5) σ (X) + [x E (X)] 1 ) ( + 3 x + b ) ] 3 S ( (x t) ) ( f (t) 1 b where S ( ) is s given by (3.4). Tking the modulus of (3.5) gives (3.6) P V (x) S ( (x t) ) ( f (t) 1 ) dt b. ) dt, Observe tht under different ssumptions with regrd to the norms of the p.d.f. f (x) we my obtin vriety of bounds. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 0 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
For f L [, b] then (3.7) P V (x) f ( ) 1 ( b S (x t) ) dt. Now, let (3.8) S ( (x t) ) (t x) ν (t X ) (t X + ), where (3.9) nd ν M ( (x ) ) (x )3 + (b x) 3 3 (b ) 1 ( ) ( b + x + b ), 3 (3.30) X x ν, X + x + ν. Then, (3.31) H (t) S ( (x t) ) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck [(t x) ν ] dt (t x)3 3 ν t + k Pge 1 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
nd so from (3.31) nd using (3.8) - (3.9) gives, (3.3) ( S (x t) ) dt H (X ) H () [H (X + ) H (X )] + [H (b) H (X + )] [H (X ) H (X + )] + H (b) H () } { ν3 3 ν X ν3 3 + ν X + (b x)3 + ν (x )3 b + + ν 3 3 [ν 3 3 ] ν3 + (b x)3 + (x ) 3 ν (b ) 3 8 3 ν3. Thus, substituting into (3.7), (3.6) nd using (3.9) redily produces the result (3.1) nd the theorem is proved. Remrk 3.. Other bounds my be obtined for f L p [, b], p 1 however obtining explicit expressions for these bounds is somewht intricte nd will not be considered further here. They involve the clcultion of sup t [,b] for f L 1 [, b] nd (t x) ν mx { (x ) ν, ν, (b x) ν } ( (t x) ν q dt ) 1 q Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
for f L p [, b], 1 p + 1 q 1, p > 1, where ν is given by (3.9). Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
4. Some Inequlities for Absolutely Continuous P.D.F s We strt with the following lemm which is interesting in itself. Lemm 4.1. Let X be rndom vrible whose probbility density function f : [, b] R + is bsolutely continuous on [, b]. Then we hve the identity (4.1) σ (X) + [E (X) x] (b ) 1 + 1 b + ( x + b where the kernel p : [, b] R is given by s, if s t b, p (t, s) : s b, if t < s b, for ll x [, b]. Proof. We use the identity (see (.10)) (4.) σ (X) + [E (X) x] for ll x [, b]. ) (t x) p (t, s) f (s) dsdt, (x t) f (t) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 4 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
On the other hnd, we know tht (see for exmple [] for simple proof using integrtion by prts) (4.3) f (t) 1 b f (s) ds + 1 b p (t, s) f (s) ds for ll t [, b]. Substituting (4.3) in (4.) we obtin (4.4) σ (X) + [E (X) x] [ 1 b (t x) f (s) ds + 1 ] p (t, s) f (s) ds dt b b 1 b 1 [ (x ) 3 + (b x) 3] 3 + 1 (t x) p (t, s) f (s) dsdt. b Tking into ccount the fct tht 1 [ (x ) 3 + (b x) 3] 3 (b ) 1 then, by (4.4) we deduce the desired result (4.1). + ( x + b ), x [, b], The following inequlity for P.D.F.s which re bsolutely continuous nd hve the derivtives essentilly bounded holds. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 5 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Theorem 4.. If f : [, b] R + is bsolutely continuous on [, b] nd f L [, b], i.e., f : ess sup f (t) <, then we hve the inequlity: t [,b] ( (4.5) σ (X) + [E (X) x] (b ) x + b ) 1 for ll x [, b]. (b ) 3 [ (b ) 10 + ( x + b ) ] f Proof. Using Lemm 4.1, we hve ( σ (X) + [E (X) x] (b ) x + b ) 1 1 b (t x) p (t, s) f (s) dsdt 1 b f b (t x) p (t, s) f (s) dsdt (t x) p (t, s) dsdt. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 6 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
We hve I : (t x) p (t, s) dsdt [ t (t x) (s ) ds + t [ ] (t x) (t ) + (b t) dt 1 [ (t x) (t ) dt + I + I b. Let A x, B b x then I 0 (t x) (t ) dt (b )3 3 ] (b s) ds dt ] (t x) (b t) dt ( u Au + A ) u du [A 3 A (b ) + 35 ] (b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 7 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
nd Now, I + I b I b 0 (b )3 3 (b )3 3 (b )3 3 (t x) (b t) dt (b )3 3 nd the theorem is proved. ( u Bu + B ) u du [B 3 B (b ) + 35 ] (b ) [ A + B 3 4 (A + B) (b ) + 3 ] 5 (b ) [ (b [ (b ) 10 ) ( + x + b + ( x + b ) 3 ) ] ] (b ) 0 The best inequlity we cn get from (4.5) is embodied in the following corollry. Corollry 4.3. If f is s in Theorem 4., then we hve [ (4.6) σ (X) + E (X) + b ] (b ) (b )4 f 1 30. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 8 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
We now nlyze the cse where f is Lebesgue p integrble mpping with p (1, ). Remrk 4.1. The results of Theorem 4. my be compred with those of Theorem 3.4. It my be shown tht both bounds re convex nd symmetric bout x +b. Further, the bound given by the premture Chebychev pproch, nmely from (3.1)-(3.13) is tighter thn tht obtined by the current pproch (4.5) which my be shown from the following. Let these bounds be described by B p nd B c so tht, neglecting the common terms nd where B p b 15 B c Y [ (b (b ) 100 ) + 15Y ] 1 + Y, ( x + b ). It my be shown through some strightforwrd lgebr tht B c B p > 0 for ll x [, b] so tht B c > B p. The current development does however hve the dvntge tht the identity (4.1) is stisfied, thus llowing bounds for L p [, b], p 1 rther thn the infinity norm. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 9 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Theorem 4.4. If f : [, b] R + is bsolutely continuous on [, b] nd f L p, i.e., ( ) 1 f p : f (t) p p dt <, p (1, ) then we hve the inequlity (4.7) σ (X) + [E (X) x] f p (b ) 1 p (q + 1) 1 q (b ) 1 [ (x ) 3q+ B ( x + b ( b + (b x) 3q+ B ) ), q + 1, q + x ( )] b, q + 1, q + b x for ll x [, b], when 1 + 1 1 nd B (,, ) is the qusi incomplete Euler s p q Bet mpping: B (z; α, β) : z 0 (u 1) α 1 u β 1 du, α, β > 0, z 1. Proof. Using Lemm 4.1, we hve, s in Theorem 4., tht ( (4.8) σ (X) + [E (X) x] (b ) x + b ) 1 1 b (t x) p (t, s) f (s) dsdt. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 30 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Using Hölder s integrl inequlity for double integrls, we hve (4.9) (t x) p (t, s) f (s) dsdt ( (b ) 1 p f p ( ) 1 ( f (s) p p b dsdt where p > 1, 1 p + 1 q 1. We hve to compute the integrl ) 1 (t x) q p (t, s) q q dsdt (t x) q p (t, s) q dsdt ) 1 q, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd (4.10) D : 1 q + 1 (t x) q p (t, s) q dsdt ] (b s) q ds [ t (t x) q (s ) q ds + t [ ] (t x) q (t ) q+1 + (b t) q+1 dt q + 1 [ (t x) q (t ) q+1 dt + ] (t x) q (b t) q+1 dt. dt Title Pge Go Bck Pge 31 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Define (4.11) E : (t x) q (t ) q+1 dt. If we consider the chnge of vrible t (1 u) +ux, we hve t implies u 0 nd t b implies u b, dt (x ) du nd then x (4.1) E Define x (4.13) F : 0 [(1 u) + ux x] q [(1 u) + ux ] (x ) du (x ) 3q+ x (u 1) q u q+1 du 0( ) b (x ) 3q+ B, q + 1, q +. x (t x) q (b t) q+1 dt. If we consider the chnge of vrible t (1 v) b + vx, we hve t b implies Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
v 0, nd t implies v b, dt (x b) dv nd then b x (4.14) F 0 b b x [(1 v) b + vx x] q [b (1 v) b vx] q+1 (x b) dv (b x) 3q+ b x (v 1) q v q+1 dv 0( ) b (b x) 3q+ B, q + 1, q +. b x Now, using the inequlities (4.8)-(4.9) nd the reltions (4.10)-(4.14), since D 1 (E + F ), we deduce the desired estimte (4.7). q+1 The following corollry is nturl to be considered. Corollry 4.5. Let f be s in Theorem 4.4. Then, we hve the inequlity: (4.15) σ (X) + [ E (X) + b ] f p (b ) + 3 q (q + 1) 1 q 3+ q (b ) 1 [B (q + 1, q + 1) + Ψ (q + 1, q + )] 1 q, where 1 + 1 1, p > 1 nd B (, ) is Euler s Bet mpping nd Ψ (α, β) : p q 1 0 uα 1 (u + 1) β 1 du, α, β > 0. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 33 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Proof. In (4.7) put x +b B (, q + 1, q + ). The left side is cler. Now 0 1 The right hnd side of (4.7) is thus: ( f b ) 3q+ q p (b ) 1 1 p (q + 1) q 0 (u 1) q u q+1 du (u 1) q u q+1 du + 1 (u 1) q u q+1 du B (q + 1, q + ) + Ψ (q + 1, q + ). [B (q + 1, q + ) + Ψ (q + 1, q + )] 1 q f p (b ) + 3 q (q + 1) 1 q 3+ q nd the corollry is proved. [B (q + 1, q + ) + Ψ (q + 1, q + )] 1 q Finlly, if f is bsolutely continuous, f L 1 [, b] nd f 1 f (t) dt, then we cn stte the following theorem. Theorem 4.6. If the p.d.f., f : [, b] R + is bsolutely continuous on [, b], then ( (4.16) σ (X) + [E (X) x] (b ) x + b ) 1 [ 1 f 1 (b ) (b ) + x + b ] Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 34 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
for ll x [, b]. Proof. As bove, we cn stte tht σ (X) + [E (X) x] (b ) 1 where 1 b ( x + b ) (t x) p (t, s) f (s) dsdt [ sup (t x) p (t, s) ] 1 (t,s) [,b] b f 1 G G : nd the theorem is proved. [ sup (t x) p (t, s) ] (t,s) [,b] (b ) sup (t x) t [,b] (b ) [mx (x, b x)] [ 1 (b ) (b ) + x + b ], f (s) dsdt It is cler tht the best inequlity we cn get from (4.16) is the one when x +b, giving the following corollry. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 35 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
Corollry 4.7. With the ssumptions of Theorem 4.6, we hve: [ (4.17) σ (X) + E (X) + b ] (b ) (b )3 f 1 4 1. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 36 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
References [1] P. CERONE AND S.S. DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res. Rep. Coll., (4) (1999), Article 8. [ONLINE] http://rgmi.vu.edu.u/vn4.html. [] P. CERONE AND S.S. DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Accepted for publiction in Anlytic- Computtionl Methods in Applied Mthemtics, G.A. Anstssiou (Ed), CRC Press, New York (000), 65 134. [3] P. CERONE AND S.S. DRAGOMIR, Midpoint type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, G.A. Anstssiou (Ed), CRC Press, New York (000), 135 00. [4] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin Mth. J., 15(1) (1999), 1 9. Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 4, 1998. [ONLINE] http://rgmi.vu.edu.u/v1n1.html. [5] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequlity of Ostrowski-Grüss type for twice differentible mppings nd pplictions, Kyungpook Mth. J., 39() (1999), 331 341. Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 8, 1998. [ONLINE] http://rgmi.vu.edu.u/v1n.html. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 37 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
[6] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p (, b) nd pplictions, Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 5. [ONLINE] http://rgmi.vu.edu.u/v1n1.html. [7] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, On Ostrowski type for mppings whose second derivtives belong to L 1 (, b) nd pplictions, Honm Mth. J., 1(1) (1999), 17 137. Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 7. [ONLINE] http://rgmi.vu.edu.u/v1n.html. [8] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth., 3() (1999), 697 71. Preprint. RGMIA Res. Rep Coll., 1() (1998), 51 66. [ONLINE] http://rgmi.vu.edu.u/v1n.html [9] P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS AND J. SUNDE, A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions, Demonstrtio Mth., 33(4) (000), 719 736. RGMIA Res. Rep. Coll., (5) (1999), Article 7. [ONLINE] http://rgmi.vu.edu.u/vn5.html. [10] S.S. DRAGOMIR, Grüss type integrl inequlity for mppings of r- Hölder s type nd pplictions for trpezoid formul, Tmkng J. Mth., 31(1) (000), 43 47. [11] S.S. DRAGOMIR, A Tylor like formul nd ppliction in numericl integrtion, submitted. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 38 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
[1] S.S. DRAGOMIR, Grüss inequlity in inner product spces, Austrl. Mth. Soc. Gz., 6() (1999), 66 70. [13] S.S. DRAGOMIR, New estimtion of the reminder in Tylor s formul using Grüss type inequlities nd pplictions, Mth. Inequl. Appl., () (1999), 183 194. [14] S.S. DRAGOMIR, Some integrl inequlities of Grüss type, Indin J. of Pure nd Appl. Mth., 31(4) (000), 397 415. [15] S.S. DRAGOMIR AND N. S. BARNETT, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, J. Indin Mth. Soc., 66(1-4) (1999), 37 45. Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 9. [ONLINE] http://rgmi.vu.edu.u/v1n.html [16] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remrks on the midpoint rule in numericl integrtion, Studi Mth. Bbeş-Bolyi Univ., (in press). [17] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J. Pure Appl. Mth., 31(5) (000), 475 494. Preprint: RGMIA Res. Rep. Coll., (5) (1999), Article 1. [ONLINE] http://rgmi.vu.edu.u/vn5.html. [18] S.S. DRAGOMIR, Y.J. CHO AND S.S. KIM, Some remrks on the Milovnović-Pečrić Inequlity nd in Applictions for specil mens nd numericl integrtion, Tmkng J. Mth., 30(3) (1999), 03 11. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 39 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
[19] S.S. DRAGOMIR AND A. McANDREW, On Trpezoid inequlity vi Grüss type result nd pplictions, Tmkng J. Mth., 31(3) (000), 193 01. RGMIA Res. Rep. Coll., () (1999), Article 6. [ONLINE] http://rgmi.vu.edu.u/vn.html. [0] S.S. DRAGOMIR, J.E. PEČARIĆ AND S. WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlity for monotonic mppings nd pplictions, Mth. nd Comp. Modelling, 31 (000), 61 70. Preprint: RGMIA Res. Rep. Coll., (4) (1999), Article 3. [ONLINE] http://rgmi.vu.edu.u/vn4.html. [1] S.S. DRAGOMIR AND A. SOFO, An integrl inequlity for twice differentible mppings nd pplictions, Preprint: RGMIA Res. Rep. Coll., () (1999), Article 9. [ONLINE] http://rgmi.vu.edu.u/vn.html. [] S.S. DRAGOMIR AND S. WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Comput. Mth. Appl., 33 (1997), 15. [3] M. MATIĆ, J.E. PEČARIĆ AND N. UJEVIĆ, On New estimtion of the reminder in Generlised Tylor s Formul, Mth. Inequl. Appl., (3) (1999), 343 361. [4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, 1993. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 40 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u
[5] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, 1994. [6] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Prtil Orderings, nd Sttisticl Applictions, Acdemic Press, 199. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 41 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001 http://jipm.vu.edu.u