Journl of Inequlities in Pure nd Applied Mthemtics http://jipmvueduu/ Volume, Issue, Article, 00 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 SCHOOL OF COMMUNICATIONS AND INFORMATICS, PO BOX 448, MELBOURNE CITY MC 800, VICTORIA, AUSTRALIA neil@mtildvueduu pc@mtildvueduu sever@mtildvueduu johnr@mtildvueduu Received 7 Jnury, 000; ccepted 6 June, 000 Communicted by CEM Perce ABSTRACT Some inequlities for the dispersion of rndom vrible whose pdf is defined on finite intervl nd pplictions re given Key words nd phrses: Rndom vrible, Expecttion, Vrince, Dispersion, Grüss Inequlity, Chebychev s Inequlity, Lupş Inequlity 000 Mthemtics Subject Clssifiction 60E5, 6D5 INTRODUCTION In this note we obtin some inequlities 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 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 INEQUALITIES FOR DISPERSION Let f :, b R R + be the pdf of the rndom vrible X nd E X) : ISSN electronic): 44-5756 c 00 Victori University All rights reserved 00-99 tf t) dt
NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS its expecttion nd σ X) t E X)) f t) dt t f t) dt E X) its dispersion or stndrd devition The following theorem holds Theorem With the bove ssumptions, we hve b ) f 6, provided f L,, b ; + b ) q ) 0 σ X) f p, provided f L p, b Proof Korkine s identity 4, is ) p t) dt q+)q+) q b ) p t) g t) h t) dt nd p >, p + q ; p t) g t) dt p t) h t) dt p t) p s) g t) g s)) h t) h s)) dtds, 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 ) σ X) It is obvious tht 4) f t) f s) t s) dtds f t) f s) t s) dtds sup t,s),b f t) f s) t s) dtds b )4 f 6 nd then, by ), we obtin the first prt of ) For the second prt, we pply Hölder s integrl inequlity for double integrls to obtin ) f t) f s) t s) dtds f p t) f p p b s) dtds t s) q dtds f p b ) q+ q, q + ) q + ) where p > nd +, nd the second inequlity in ) is proved p q For the lst prt, observe tht s f t) f s) t s) dtds f t) f s) dtds sup t s) t,s),b f t) dt f t) f s) dtds b ) f s) ds ) q J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE Using finer rgument, the lst inequlity 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 inequlity: 6) 0 p t) g t) dt p t) dt ) p t) g 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 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 4 Let X be rndom vrible hving the pdf 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 b x) q+ +x ) q+ b Proof We observe tht q+ + x +b ) ) f, provided f L, b ; q f p, provided f L p, b, p >, nd p + q ; 8) nd s x t) f t) dt x xt + t ) f t) dt x xe X) + t f t) dt 9) σ X) we get, by 8) nd 9), 0) x E X) + σ X) which is of interest in itself too t f t) dt E X), x t) f t) dt, J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
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 inequlity in 7) is proved For the second inequlity, observe tht by Hölder s integrl inequlity, ) x t) f t) dt f p p b t) dt x t) q dt nd the second inequlity in 7) is estblished Finlly, observe tht, nd the theorem is proved ) q b x) q+ + x ) q+ q f p, q + x t) f t) dt sup x t) f t) dt t,b mx { x ), b x) } The following corollries re esily deduced Corollry 5 With the bove ssumptions, we hve ) 0 σ X) b ) b ) + E X) +b b EX)) q+ +EX) ) q+ q+ mx {x, b x}) b + x + b ), ) f, provided f L, b ; q f p, if f L p, b, p > nd p + q ; b + E X) +b Remrk 6 The lst inequlity in ) 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 b ) min x,b + x + b ) b x) q+ + x ) q+ min x,b q + b ) b )q+ q q + ),, J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 nd b min + x,b x + b Consequently, we cn stte the following corollry s well b Corollry 7 With the bove ssumptions, we hve the inequlity: 0 σ X) + E X) + b ) b ) f, provided f L, b ; b ) q+ f p, if f L p, b, p >, 4q+) q b ) 4 nd p + q ; Remrk 8 From the lst inequlity in ), we obtin ) 0 σ X) b E X)) E X) ) 4 b ), which is n improvement on 5) PERTURBED RESULTS USING GRÜSS TYPE INEQUALITIES In 95, 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 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, ) T h, g) Φ φ) Γ γ), 4 where ) T h, g) b h x) g x) dx b h x) dx b g x) dx nd the inequlity is shrp, in the sense tht the constnt cnnot be replced by smller 4 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 inequlity is embodied in the following theorem which ws proved in It provides shrper bound thn the bove Grüss inequlity Theorem Let h, g be integrble functions defined on, b nd let d g t) D Then ) T h, g) D d where T h, g) is s defined in ) T h, h), Theorem will now be used to provide perturbed rule involving the vrince nd men of pdf J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
6 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Perturbed Results Using Premture Inequlities In this subsection we develop some perturbed results Theorem 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 inequlity 4) P V x) : σ X) + x E X)) M m M m) 6) T h, h) b Now, b ) 45 b ) 45 b b ) x + b ) ) + 5 x + b ) Proof Applying the premture Grüss result ) by ssociting g t) with f t) nd h t) x t), gives, from )-) 5) x t) f t) dt x t) dt f t) dt b b ) M m T h, h), where from ) b x t) 4 dt x t) dt b 7) nd b x t) dt x ) + b x) b ) b b x t) 4 dt x )5 + b x) 5 5 b ) ) + x + b ) giving, for 6), x ) 5 + b x) 5 x ) + b x) 8) 45T h, h) 9 5 b b 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 ) 9) T h, h) + 5 x + b ) 45 J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 7 nd from 7) b giving 0) x t) dt A + B A + B) b x t) dt A A AB + B ) + B + b ) + x + b ) ) A B, Hence, from 5), 9) 0) nd 0), the first inequlity 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 inequlity obtinble 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 inequlity 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 5 Let the conditions on Theorem be stisfied Further, suppose tht f is differentible nd is such tht f : sup f t) < t,b Then ) P V x) b f I x), 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 inequlity 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 4) T h, g) b ) 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 J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
8 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Theorem 6 Let the conditions of Theorem 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) b ) π h g, ) k : k t) for k L, b) b Mtić, Pečrić nd Ujević further show tht 6) T h, g) b π 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 Inequlities Involving the Vrince Let 7) S h x)) h x) M h) where 8) M h) b Then from ), h u) du 9) T h, g) M hg) M h) M g) Drgomir nd McAndrew 9 hve shown, tht 0) T h, g) T S h), S g)) nd proceeded to obtin bounds for trpezoidl rule Identity 0) is now pplied to obtin bounds for the vrince Theorem 7 Let X be rndom vrible hving the pdf f :, b R R + Then for ny x, b the following inequlity holds, nmely, ) P V x) 8 ν x) f ) if f L, b, b where P V x) is s defined by the left hnd side of 4), nd ν ν x) b Proof Using identity 0), ssocite with h ), x ) nd f ) with g ) Then ) + ) x +b ) x t) f t) dt M x ) ) x t) M x ) ) f t) dt, b J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 9 where from 8), M x ) ) b x t) dt x ) + b x) b ) nd so ) M x ) ) ) b + x + b ) Further, from 7), nd so, on using ) 4) S x ) ) x t) S x ) ) x t) M x ) ) b ) x + b ) Now, from ) nd using 0), ) nd 4), the following identity is obtined b 5) σ X) + x E X) ) + x + b ) where S ) is s given by 4) Tking the modulus of 5) gives 6) P V x) S x t) ) f t) ) dt b S x t) ) f t) b ) dt, Observe tht under different ssumptions with regrd to the norms of the pdf f x) we my obtin vriety of bounds For f L, b then 7) P V x) f ) b S x t) ) dt Now, let 8) S x t) ) t x) ν t X ) t X + ), where 9) ν M x ) ) x ) + b x) b ) nd b ) + x + b ), 0) X x ν, X + x + ν Then, ) H t) S x t) ) t dt x) ν dt t x) ν t + k J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
0 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS 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 x) ν X + + ν b + ν ν + b x) + x ) ν b ) x ) + ν 8 ν Thus, substituting into 7), 6) nd using 9) redily produces the result ) nd the theorem is proved Remrk 8 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,b for f L, b nd t x) ν mx { x ) ν, ν, b x) ν } t x) ν q dt for f L p, b, p + q, p >, where ν is given by 9) 4 SOME INEQUALITIES FOR ABSOLUTELY CONTINUOUS PDFS We strt with the following lemm which is interesting in itself Lemm 4 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 ) q 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 for ll x, b x t) f t) dt t x) p t, s) f s) dsdt, J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE On the other hnd, we know tht see for exmple for simple proof using integrtion by prts) 4) f t) b for ll t, b Substituting 4) in 4) we obtin 44) σ X) + E X) x t x) b b Tking into ccount the fct tht f s) ds + b f s) ds + b x ) + b x) + b x ) + b x) b ) + p t, s) f s) ds p t, s) f s) ds dt t x) p t, s) f s) dsdt x + b ), x, b, then, by 44) we deduce the desired result 4) The following inequlity for PDFs which re bsolutely continuous nd hve the derivtives essentilly bounded holds Theorem 4 If f :, b R + is bsolutely continuous on, b nd f L, b, ie, f : ess sup f t) <, then we hve the inequlity: t,b 45) σ X) + E X) x for ll x, b Proof Using Lemm 4, we hve σ X) + E X) x b ) b ) x + b b b f b b ) ) b ) 0 x + b ) + x + b ) f t x) p t, s) f s) dsdt t x) p t, s) f s) dsdt t x) p t, s) dsdt J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS We hve I : I + I b Let A x, B b x then nd Now, I + I b t x) p t, s) dsdt t t x) s ) ds + t b t x) t ) + b t) dt t x) t ) dt + I I b nd the theorem is proved 0 t x) t ) dt b ) 0 b ) b ) b ) u Au + A ) u du b s) ds dt t x) b t) dt A A b ) + 5 b ) t x) b t) dt b ) u Bu + B ) u du B B b ) + 5 b ) A + B 4 A + B) b ) + 5 b ) b ) + x + b ) b ) 0 b ) + x + b ) 0 The best inequlity we cn get from 45) is embodied in the following corollry Corollry 4 If f is s in Theorem 4, then we hve 46) σ X) + E X) + b b ) b )4 f 0 We now nlyze the cse where f is Lebesgue p integrble mpping with p, ) J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE Remrk 44 The results of Theorem 4 my be compred with those of Theorem 5 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 nd where B p b 5 B c Y b b ) 00 ) + 5Y + 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) is stisfied, thus llowing bounds for L p, b, p rther thn the infinity norm Theorem 45 If f :, b R + is bsolutely continuous on, b nd f L p, ie, then we hve the inequlity 47) f p : ) f t) p p dt σ X) + E X) x b ) f p b ) p q + ) q x + b x ) q+ B <, p, ) ) b, q +, q + x + b x) q+ B ) ) b, q +, q + b x for ll x, b, when p + q nd B,, ) is the qusi incomplete Euler s Bet mpping: B z; α, β) : z 0 u ) α u β du, α, β > 0, z Proof Using Lemm 4, we hve, s in Theorem 4, tht 48) σ X) + E X) x b ) x + b ) b t x) p t, s) f s) dsdt J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
4 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Using Hölder s integrl inequlity for double integrls, we hve 49) t x) p t, s) f s) dsdt where p >, + p q We hve to compute the integrl 40) Define D : q + 4) E : b ) p f p t x) q p t, s) q dsdt ) f s) p p b dsdt t t x) q s ) q ds + t t x) q t ) q+ + b t) q+ dt q + t x) q t ) q+ dt + ) t x) q p t, s) q q dsdt t x) q p t, s) q dsdt b s) q ds t x) q t ) q+ dt dt ) q t x) q b t) q+ dt If we consider the chnge of vrible t u) + ux, we hve t implies u 0 nd t b implies u b, dt x ) du nd then x 4) Define E x 0 u) + ux x q u) + ux x ) du x ) q+ x u ) q u q+ du 0 ) b x ) q+ B, q +, q + x 4) F : t x) q b t) q+ dt If we consider the chnge of vrible t v) b + vx, we hve t b implies v 0, nd t implies v b, dt x b) dv nd then 44) F b x 0 b b x v) b + vx x q b v) b vx q+ x b) dv b x) q+ b x v ) q v q+ dv 0 ) b b x) q+ B, q +, q + b x, J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 Now, using the inequlities 48)-49) nd the reltions 40)-44), since D E + F ), q+ we deduce the desired estimte 47) The following corollry is nturl to be considered Corollry 46 Let f be s in Theorem 45 Then, we hve the inequlity: 45) σ X) + E X) + b b ) f p b ) + q q + ) q + q B q +, q + ) + Ψ q +, q + ) q, where p + q, 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 Now B, q +, q + ) The right hnd side of 47) is thus: f b ) q+ q p b ) p q + ) q 0 0 u ) q u q+ du u ) q u q+ du + u ) q u q+ du B q +, q + ) + Ψ q +, q + ) B q +, q + ) + Ψ q +, q + ) q f p b ) + q q + ) q + q B q +, q + ) + Ψ q +, q + ) q nd the corollry is proved Finlly, if f is bsolutely continuous, f L, b nd f f t) dt, then we cn stte the following theorem Theorem 47 If the pdf, f :, b R + is bsolutely continuous on, b, then 46) σ X) + E X) x for ll x, b b ) x + b ) f b ) b ) + x + b J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
6 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Proof As bove, we cn stte tht σ X) + E X) x b ) where nd the theorem is proved b x + b ) t x) p t, s) f s) dsdt sup t x) p t, s) t,s),b b f G G : sup t,s),b t x) p t, s) b ) sup t x) t,b b ) mx x, b x) b ) b ) + x + b, f s) dsdt It is cler tht the best inequlity we cn get from 46) is the one when x +b, giving the following corollry Corollry 48 With the ssumptions of Theorem 47, we hve: 47) σ X) + E X) + b b ) b ) f 4 REFERENCES P CERONE AND SS DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res Rep Coll, 4) 999), Article 8 ONLINE http://rgmivueduu/vn4html P CERONE AND SS DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), 65 4 P CERONE AND SS DRAGOMIR, Midpoint type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), 5 00 4 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin Mth J, 5) 999), 9 Preprint RGMIA Res Rep Coll, ) 998), Article 4, 998 ONLINE http://rgmivueduu/vnhtml 5 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, An inequlity of Ostrowski- Grüss type for twice differentible mppings nd pplictions, Kyungpook Mth J, 9) 999), 4 Preprint RGMIA Res Rep Coll, ) 998), Article 8, 998 ONLINE http://rgmivueduu/vnhtml J Inequl Pure nd Appl Mth, ) Art, 00 http://jipmvueduu/
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