SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL

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1 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Abstrct Some ineulities for the dispersion of rndom vrible whose pdf is defined on finite intervl nd pplictions re given Introduction In this note we obtin some ineulities for the dispersion of continuous rndom vrible X hving the probbility density function pdf) f defined on finite intervl b Tools used include: Korkine s identity which plys centrl role in the proof of Chebychev s integrl ineulity for synchronous mppings 4 Hölder s weighted ineulity 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 ineulities In Section 4 results from n identity involving double integrl re obtined for vriety of norms Some Ineulities for Dispersion Let f : b R R + be the pdf of the rndom vrible X nd its expecttion nd E X) : tf t) dt σ X) t E X)) f t) dt t f t) dt E X) its dispersion or stndrd devition The following theorem holds Dte: November Mthemtics Subject Clssifiction Primry 60E5; Secondry 6D5 Key words nd phrses Rndom vrible Expecttion Vrince Dispertion Grüss Ineulity Chebychev s Ineulity Lupş Ineulity

2 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Theorem With the bove ssumptions we hve b ) 6 f provided f L b ; + b ) ) 0 σ X) f p provided f L p b +)+) b ) Proof Korkine s identity 4 is ) p t) dt p t) g t) h t) dt nd p > p + ; p t) g t) dt p t) p s) g t) g s)) h t) h s)) dtds p t) h t) dt 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 ) It is obvious tht 4) σ X) f t) f s) t s) dtds f t) f s) t s) dtds sup f t) f s) ts) b b )4 6 f t s) dtds nd then by ) we obtin the first prt of ) For the second prt we pply Hölder s integrl ineulity for double integrls to obtin f p f t) f s) t s) dtds f p t) f p s) dtds ) b ) + + ) + ) p t s) dtds where p > nd p + nd the second ineulity in ) is proved For the lst prt observe tht f t) f s) t s) dtds sup ts) b t s) b ) ) f t) f s) dtds

3 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE s f t) f s) dtds f t) dt f s) ds Using finer rgument the lst ineulity in ) cn be improved s follows Theorem Under the bove ssumptions we hve 5) 0 σ X) b ) Proof We use the following Grüss type ineulity: 0 p t) ) g t) dt p t) g t) dt 6) 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 ineulity see 9 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 The sme conclusion cn be obtined for the choice p t) f t) nd g t) t t b The following result holds Theorem Let X be rndom vrible hving the pdf given by f : b R R + Then for ny x b we hve the ineulity: 7) σ X) + x E X)) b ) b ) + x +b b x) + +x ) + b + + x +b Proof We observe tht 8) nd s 9) we get by 8) nd 9) 0) ) x t) f t) dt σ X) ) f provided f L b ; f p provided f L p b p > nd p + ; x xt + t ) f t) dt x xe X) + x E X) + σ X) which is of interest in itself too t f t) dt E X) x t) f t) dt t f t) dt

4 4 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS We observe tht x t) f t) dt ess sup f t) t b x t) dt b x) + x ) f b ) b ) f + x + b ) nd the first ineulity in 7) is proved For the second ineulity observe tht by Hölder s integrl ineulity x t) f t) dt ) b p f p t) dt nd the second ineulity in 7) is estblished Finlly observe tht nd the theorem is proved x t) dt ) b x) + + x ) + f p + x t) f t) dt sup t b The following corollries re esily deduced x t) f t) dt mx {x ) b x) } mx {x b x}) b + x + b ) Corollry With the bove ssumptions we hve b ) b ) + E X) +b ) f ) 0 σ X) provided f L b ; b EX)) + +EX) ) + + f p if f L p b p > nd p + ; + E X) +b b Remrk The lst ineulity in ) is worse thn the ineulity 5) obtined by techniue bsed on Grüss ineulity

5 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 The best ineulity we cn get from 7) is tht one for which x +b nd this pplies for ll the bounds s nd min x b b ) + x + b ) b x) + + x ) + min x b + b min x b + x + b b ) b )+ + ) b Conseuently we cn stte the following corollry s well Corollry With the bove ssumptions we hve the ineulity: 0 σ X) + E X) + b ) b ) f provided f L b ; b ) + f p if f L p b p > 4+) b ) nd p + ; Remrk from the lst ineulity in ) we obtin ) 0 σ X) b E X)) E X) ) 4 b ) which is n improvement on 5) Perturbed Results Using Grüss Type ineulities In 95 G Grüss see for exmple 6) proved the following integrl ineulity which gives n pproximtion for the integrl of product in terms of the product of the integrls Theorem 4 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 ) where ) T h g) b T h g) Φ φ) Γ γ) 4 h x) g x) dx b h x) dx b g x) dx nd the ineulity is shrp in the sense tht the constnt 4 cnnot be replced by smller one For simple proof of this s well s for extensions generlistions discrete vrints nd other ssocited mteril see 5 nd - where further references re given A premture Grüss ineulity is embodied in the following theorem which ws proved in It provides shrper bound thn the bove Grüss ineulity

6 6 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Theorem 5 Let h g be integrble functions defined on b nd let d g t) D Then ) T h g) D d T h h) where T h g) is s defined in ) Theorem 5 will now be used to provide perturbed rule involving the vrince nd men of pdf Perturbed Results Using Premture Ineulities In this subsection we develop some perturbed results Theorem 6 Let X be rndom vrible hving the pdf given by f : b R R + Then for ny x b nd m f x) M we hve the ineulity P V x) : σ X) + x E X)) b ) x + b ) 4) b M m ) b ) + 5 x + b ) 45 M m) b ) 45 Proof Applying the premture Grüss result ) by ssociting g t) with f t) nd h t) x t) gives from )-) x t) f t) dt x t) 5) dt f t) dt b where from ) 6) Now 7) nd b ) M m T h h) b b T h h) x t) 4 dt b x t) dt x t) dt x ) + b x) b ) ) b + x + b b x t) 4 dt x )5 + b x) 5 5 b ) giving from 6) x ) 5 + b x) 5 x ) + b x) 8) 45T h h) 9 5 b b )

7 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 7 Let A x nd B b x in 8) to give A 5 + B 5 ) A + B 45T h h) 9 5 A + B A + B 9 A 4 A B + A B AB + B 4 5 A AB + B ) 4A 7AB + 4B ) A + B) A ) ) + B A B + 5 A + B) Using the fcts tht A + B b nd A B x + b) gives b ) b ) T h h) + 5 x + b ) 9) 45 nd from 7) giving 0) b b x t) dt A + B A + B) A AB + B A ) ) + B A B + x t) dt b ) + x + b ) Hence from 5) 9) 0) nd 0) the first ineulity in 4) results The corsest uniform bound is obtined by tking x t either end point Thus the theorem is completely proved Remrk 4 The best ineulity otinble from 4) is t x +b giving σ X) + E X) + b b ) M m b ) ) 5 The result ) is tighter bound thn tht obtined in the first ineulity of ) since 0 < M m < f For symmetric pdf E X) +b nd so the bove results would give bounds on the vrince The following results hold if the pdf f x) is differentible tht is for f x) bsolutely continuous Theorem 7 Let the conditions on Theorem 4 be stisfied Further suppose tht f is differentible nd is such tht Then ) f : sup f t) < t b P V x) b f I x)

8 8 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS where P V x) is given by the left hnd side of 4) nd b ) b ) I x) + 5 x + b ) ) 45 Proof Let h g : b R be bsolutely continuous nd h g be bounded Then Chebychev s ineulity holds see ) T h g) b ) sup h t) sup g t) t b t b Mtić Pečrić nd Ujević using premture Grüss type rgument proved tht b ) 4) T h g) sup g t) T h h) t b Associting f ) with g ) nd x ) with h ) in ) gives from 5) nd 9) I x) b ) T h h) which simplifies to ) nd the theorem is proved Theorem 8 Let the conditions of Theorem 6 be stisfied Further suppose tht f is loclly bsolutely continuous on b) nd let f L b) Then 5) P V x) b π f I x) where P V x) is the left hnd side of 4) nd I x) is s given in ) Proof The following result ws obtined by Lupş see ) For h g : b) R loclly bsolutely continuous on b) nd h g L b) then where T h g) k : b b ) π h g k t) ) Mtić Pečrić nd Ujević further show tht 6) T h g) b π for k L b) g T h h) Associting f ) with g ) nd x ) with h in 6) gives 5) where I x) is s found in ) since from 5) nd 9) I x) b ) T h h) Alternte Grüss Type Results for Ineulities Involving the Vrince Let 7) where 8) Then from ) 9) S h x)) h x) M h) M h) b h u) du T h g) M hg) M h) M g)

9 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 9 Drgomir nd McAndrew 9 hve shown tht 0) T h g) T S h) S g)) nd proceeded to obtin bounds for trpezoidl rule pplied to obtin bounds for the vrince Identity 0) is now Theorem 9 Let X be rndom vrible hving the pdf f : b R R + Then for ny x b the following ineulity holds nmely P V x) 8 ν x) f ) ) b if f L b where P V x) is s defined by the left hnd side of 4) nd ν ν x) b ) + ) x +b Proof Using identity 0) ssocite with h ) x ) nd f ) with g ) Then ) x t) f t) dt M x ) ) where from 8) M x ) ) b nd so ) Further from 7) x t) M x ) ) f t) dt b M x ) ) x t) dt b x ) + b x) b ) ) + x + b ) S x ) ) x t) M x ) ) nd so on using ) S x ) ) x t) ) b x + b ) 4) Now from ) nd using 0) ) nd 4) the following identity is obtined b σ X) + x E X) ) + x + b ) 5) S x t) ) f t) b ) dt where S ) is s given by 4) Tking the modulus of 5) gives P V x) S x t) ) f t) ) 6) dt b Observe tht under different ssumptions with regrd to the norms of the pdf f x) we my obtin vriety of bounds

10 0 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS For f L b then P V x) f ) 7) b S Now let 8) where 9) nd 0) Then ) x t) ) dt S x t) ) t x) ν t X ) t X + ) ν M x ) ) x ) + b x) b ) ) b + x + b ) H t) X x ν X + x + ν S x t) ) dt t x) ν dt t x) ν t + k nd so from ) nd using 8) - 9) gives S x t) ) ) dt H X ) H ) H X + ) H X ) + H b) H X + ) H X ) H X + ) + H b) H ) } { ν ν X ν + b ν x) X + + ν b + ν ν + b x) + x ) ν b ) 8 ν x ) + ν Thus substituting into 7) 6) nd using 9) redily produces the result ) nd the theorem is proved Remrk 5 Other bounds my be obtined for f L p b p however obtining explicit expressions for these bounds is somewht intricte nd will not be considered further here They involve the clcultion of { } sup t x) ν x mx ) ν ν b x) ν t b for f L b nd ) b t x) ν dt for f L p b p + p > where ν is given by 9)

11 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 4 Some Ineulities for Absolutely Continuous PDFs We strt with the following lemm which is interesting in itself Lemm Let X be rndom vrible whose probbility density function f : b R + is bsolutely continuous on b Then we hve the identity 4) σ X) + E X) x b ) + x + b ) + b t x) p t s) f s) dsdt 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 0)) 4) σ X) + E X) x x t) f t) dt for ll x b On the other hnd we know tht see for exmple for simple proof using integrtion by prts) 4) f t) b f s) ds + b p t s) f s) ds for ll t b Substituting 4) in 4) we obtin 44) σ X) + E X) x b t x) b b b f s) ds + b x ) + b x) + b Tking into ccount the fct tht p t s) f s) ds dt t x) p t s) f s) dsdt x ) + b x) b ) + x + b ) x b then by 44) we deduce the desired result 4) The following ineulity for PDFs which re bsolutely continuous nd hve the derivtives essentilly bounded holds

12 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Theorem 0 If f : b R + is bsolutely continuous on b nd f L b ie f : ess sup f t) < then we hve the ineulity: t b 45) for ll x b σ X) + E X) x b ) x + b ) b ) b ) + x + b ) f 0 Proof Using Lemm we hve σ X) + E X) x b ) x + b ) t x) p t s) f s) dsdt b We hve I : b f b t x) p t s) f s) dsdt t x) p t s) dsdt t x) p t s) dsdt t t x) s ) ds + t b s) ds t x) t ) + b t) dt b b t x) t ) dt + t x) b t) dt I + I b ) Let A x B b x then dt I 0 t x) t ) dt b ) u Au + A ) u du A A b ) + 5 b )

13 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE nd Now I + I b I b nd the theorem is proved 0 t x) b t) dt b ) u Bu + B ) u du B B b ) + 5 b ) b ) A + B 4 A + B) b ) + 5 b ) b ) b ) + x + b ) b ) 0 b ) b ) + x + b ) 0 The best ineulity we cn get from 45) is embodied in the following corollry Corollry If f is s in Theorem 0 then we hve σ X) + E X) + b b ) b )4 46) f 0 We now nlyze the cse where f is Lebesgue p integrble mpping with p ) Remrk 6 The results of Theorem 0 my be compred with those of Theorem 7 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 )-) is tighter thn tht obtined by the current pproch 45) which my be shown from the following Let these bounds be described by B p nd B c so tht neglecting the common terms b B p b ) + 5Y 5 nd where B c Y b ) 00 + Y x + b ) It my be shown through some strightforwrd lgebr tht Bc Bp > 0 for ll x b so tht B c > B p The current development does however hve the dvntge tht the identity 4) is stisfied thus llowing bounds for L p b p rther thn the infinity norm

14 4 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Theorem If f : b R + is bsolutely continuous on b nd f L p ie ) b p f p : f t) p dt < p ) then we hve the ineulity σ X) + E X) x b ) 47) f p x ) + b B b ) p + ) ) + b x) + b B + + b x x + b ) + + x for ll x b when p + nd B ) is the usi incomplete Euler s Bet mpping: B z; α β) : z 0 u ) α u β du α β > 0 z Proof Using Lemm we hve s in Theorem 0 tht σ X) + E X) x b ) x + b ) 48) b t x) p t s) f s) dsdt Using Hölder s integrl ineulity for double integrls we hve 49) t x) p t s) f s) dsdt f s) p dsdt b ) p f p where p > p + We hve to compute the integrl 40) D : + ) p t x) p t s) dsdt t t x) s ) ds + t x) p t s) dsdt t x) p t s) dsdt t b s) ds ) dt ) ) t x) t ) + + b t) + dt + b t x) t ) + dt + t x) b t) + dt

15 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 Define 4) E : t x) t ) + dt If we consider the chnge of vrible t u) +ux we hve t implies u 0 nd t b implies u b x dt x ) du nd then 4) E x 0 u) + ux x u) + ux x ) du x ) + x u ) u + du 0 ) x ) + b B + + x Define 4) F : t x) b t) + dt If we consider the chnge of vrible t v) b+vx we hve t b implies v 0 nd t implies v b b x dt x b) dv nd then 44) F 0 b b x v) b + vx x b v) b vx + x b) dv b x) + b x v ) v + dv 0 ) b x) + b B + + b x Now using the ineulities 48)-49) nd the reltions 40)-44) since D E + F ) we deduce the desired estimte 47) + The following corollry is nturl to be considered Corollry 4 Let f be s in Theorem Then we hve the ineulity: σ X) + E X) + b b ) 45) f p b ) + + ) + B + + ) + Ψ + + ) where p + p > nd B ) is Euler s Bet mpping nd Ψ α β) : 0 uα u + ) β du α β > 0 Proof In 47) put x +b The left side is cler

16 6 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Now B + + ) 0 The right hnd side of 47) is thus:- f p b 0 u ) u + du u ) u + du + u ) u + du B + + ) + Ψ + + ) ) + b ) p + ) f p b ) + + ) + nd the corollry is proved B + + ) + Ψ + + ) B + + ) + Ψ + + ) Finlly s f is bsolutely continuous f L b nd f f t) dt nd we cn stte the following theorem Theorem If the pdf f : b R + is bsolutely continuous on b then σ X) + E X) x b ) x + b ) 46) f b ) b ) + x + b for ll x b Proof As bove we cn stte tht σ X) + E X) x where b sup ts) b f G b ) t x) p t s) f s) dsdt t x) p t s) b x + b ) f s) dsdt G : sup ts) b t x) p t s) b ) sup t b t x) nd the theorem is proved b ) mx x b x) b ) b ) + x + b It is cler tht the best ineulity we cn get from 46) is the one when x +b giving the following corollry

17 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 7 Corollry 5 With the ssumptions of Theorem we hve: σ X) + E X) + b b ) b ) 47) f 4 References P CERONE nd S S DRAGOMIR Three point udrture rules involving t most first derivtive submitted RGMIA Res Rep Coll 4 999) Article 8 P CERONE nd S S DRAGOMIR Trpezoidl type rules from n ineulities point of view Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics GA Anstssiou Ed) CRC Press New York P CERONE nd S S DRAGOMIR Midpoint type rules from n ineulities point of view Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics GA Anstssiou Ed) CRC Press New York 4 P CERONE S S DRAGOMIR nd J ROUMELIOTIS An ineulity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions Preprint RGMIA Res Rep Coll ) 998) Article ONLINE 5 P CERONE S S DRAGOMIR nd J ROUMELIOTIS An ineulity of Ostrowski-Grüss type for twice differentible mppings nd pplictions Preprint RGMIA Res Rep Coll ) 998) Article ONLINE 6 P CERONE S S DRAGOMIR nd J ROUMELIOTIS An Ostrowski type ineulity for mppings whose second derivtives belong to L p b) nd pplictions Preprint RGMIA Res Rep Coll ) 998) Article ONLINE 7 P CERONE S S DRAGOMIR nd J ROUMELIOTIS On Ostrowski type for mppings whose second derivtives belong to L b) nd pplictions Preprint RGMIA Res Rep Coll ) Article ONLINE 8 P CERONE S S DRAGOMIR nd J ROUMELIOTIS Some Ostrowski type ineulities for n-time differentible mppings nd pplictions Preprint RGMIA Res Rep Coll ) 998) 5-66 ONLINE 9 P CERONE SS DRAGOMIR J ROUMELIOTIS nd J SUNDE A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions RGMIA Res Rep Coll 5) Article ONLINE 0 SS DRAGOMIR Grüss type integrl ineulity for mppings of r-hölder s type nd pplictions for trpezoid formul Tmkng Journl of Mthemtics ccepted 999 SS DRAGOMIR A Tylor like formul nd ppliction in numericl integrtion submitted SS DRAGOMIR Grüss ineulity in inner product spces The Austrlin Mth Gzette 6 ) SS DRAGOMIR New estimtion of the reminder in Tylor s formul using Grüss type ineulities nd pplictions Mthemticl Ineulities nd Applictions 999) SS DRAGOMIR Some integrl ineulities of Grüss type Itlin J of Pure nd Appl Mth ccepted SS DRAGOMIR nd N S BARNETT An Ostrowski type ineulity for mppings whose second derivtives re bounded nd pplictions Preprint RGMIA Res Rep Coll ) 998) Article ONLINE 6 SS DRAGOMIR P CERONE nd A SOFO Some remrks on the midpoint rule in numericl integrtion submitted SS DRAGOMIR P CERONE nd A SOFO Some remrks on the trpezoid rule in numericl integrtion Indin J of Pure nd Appl Mth in press) 999 Preprint: RGMIA Res Rep Coll 5) Article SS DRAGOMIR YJ CHO nd SS KIM Some remrks on the Milovnović-Pečrić Ineulity nd in Applictions for specil mens nd numericl integrtion Tmkng Journl of Mthemtics ccepted SS DRAGOMIR nd A McANDREW On Trpezoid ineulity vi Grüss type result nd pplictions RGMIA Res Rep Coll 999) Article 6

18 8 NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS 0 SS DRAGOMIR JE PEČARIĆ nd S WANG The unified tretment of trpezoid Simpson nd Ostrowski type ineulity for monotonic mppings nd pplictions Preprint: RGMIA Res Rep Coll 4) Article 999 ONLINE SS DRAGOMIR nd A SOFO An integrl ineulity for twice differentible mppings nd pplictions Preprint: RGMIA Res Rep Coll ) Article ONLINE SS DRAGOMIR nd S WANG An ineulity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl udrture rules Computers Mth Applic M MATIĆ JE PEČARIĆ nd N UJEVIĆ On New estimtion of the reminder in Generlised Tylor s Formul MIA Vol No 999) DS MITRINOVIĆ JE PEČARIĆ nd AM FINK Clssicl nd New Ineulities in Anlysis Kluwer Acdemic Publishers 99 5 DS MITRINOVIĆ JE PEČARIĆ nd AM FINK Ineulities for Functions nd Their Integrls nd Derivtives Kluwer Acdemic Publishers JE PEČARIĆ F PROSCHAN nd YL TONG Convex Functions Prtil Orderings nd Sttisticl Applictions Acdemic Press 99 School of Communictions nd Informtics PO Box 448 Melbourne City MC 800 Victori Austrli URL: E-mil ddress: {neil pc sever johnr}@mtildvueduu