Some Applications of Dirac's Delta Function in Statistics for More Than One Random Variable
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1 Avalabl at htt://vaudu/aa Al Al Ma ISSN: Vol Issu Ju Prvousl Vol No Alcatos ad Ald Maatcs: A Itratoal Joural AAM So Alcatos o Drac's Dlta Fucto Statstcs or Mor Tha O Rado Varabl Satau Charabort Dartt o Maatcs Uvrst o Tas Pa Arca Wst Uvrst Drv Edburg Tas 7854 USA scharabort@utadu Rcvd Jul 4 6; acctd Dcbr 7 7 Abstract I s ar w dscuss so trstg alcatos o Drac's dlta ucto Statstcs W hav trd to td so o stg rsults to or a o varabl cas Whl dog at w artcularl coctrat o bvarat cas Kwords: Drac's Dlta ucto Rado Varabls Dstrbutos Dsts Talor's Srs Easos Mot gratg uctos Itroducto Cauch 86 was rst ad ddtl Posso 85 gav a drvato o Fourr tgral or b as o a argut volvg what w would ow rcogz as a salg orato o t assocatd w dlta ucto Ad r ar slar als o us o what ar sstall dlta uctos b Krcho Hlholtz ad Havsd But Drac was rst to us otato Th Drac dlta ucto -ucto was troducd b Paul Drac at d o 9s a ort to crat aatcal tools or dvlot o quatu ld or H rrrd to t as a ror ucto Drac 9 Latr 947 Laurt Schwartz gav t a or rgorous aatcal dto as a lar uctoal o sac o tst uctos D st o all ral-valud tl drtabl uctos w coact suort such at or a gv ucto D valu o uctoal s gv b rort b blow Ths s calld stg or salg rort o dlta ucto Sc dlta ucto s ot rall a ucto classcal ss o should ot cosdr valu o dlta ucto at Hc doa o dlta ucto s D ad ts valu or D ad a gv s Khur 4 studd so trstg alcatos o dlta ucto statstcs H al studd uvarat cass v ough h dd gv so trstg als or ultvarat cas W shall stud so or alcatos ultvarat scaro s wor Ths ght hl utur rsarchrs 4
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3 44 Satau Charabort H or < or dh Th gralzd drvatv o H s As a rsult w gt a scal d cas o orula or ordr drvatv tod abov: d! Drvatvs o dlta ucto bvarat cas Followg Sachv ad Woczs 997 Khur 4 rovdd tdd dto o dlta ucto to -dsoal Euclda sac But w shall al coctrat o bvarat cas As uvarat cas w ca wrt dow slar rorts or bvarat cas as wll I bvarat cas So w assu to b a cotuous ucto so ghborhood o w ca wrt R R dd whr R s ral l Now or s ucto all ts artal drvatvs u to ordr ar cotuous abovtod ghborhood o dd C R R whr C s ubr o cobatos o out o obcts s gralzd ordr drvatv o I gral -dsoal cas b usg ducto o t ca b show at d d!!!
4 AAM: Itr J Vol Issu Ju 8 [Prvousl Vol No ] 45 whr ad s a ucto o varabls al 4 Us o Dlta Fucto to Obta Dscrt Probablt Dstrbutos I s a dscrt rado varabl at assus valus a a w robablts rsctvl such at robablt ass ucto o ca b rrstd as a Now lt us cosdr two dscrt rado varabls ad Y whch assu valus a a ad b b rsctvl ad ot robablt P a Y b s gv b or ad so at ot robablt ass ucto s gv b a b Slarl o ca wrt dow ot robablt dstrbuto o a t ubr o rado varabls trs dlta uctos as ollows: Suos w hav rado varabls w tag valus a or w robablt Th ot robablt ass ucto s P a a As a al w a cosdr stuato o ultoal dstrbutos Lt ollow ultoal dstrbuto w aratrs Th!!! P whr add u to ad add u to I trs o dlta ucto ot robablt ass ucto s P!!!!
5 46 Satau Charabort W ca also cosdr codtoal robablts ad o rssg trs o - ucto Lt us go bac to al o two dscrt rado varabls ad Y whr tas valus a a a ad Y tas valus b b b Th codtoal robablt o Y gv s gv b PY PY P a b a 5 Dsts o Trasoratos o Rado Varabls Usg -ucto I s a cotuous rado varabl w a dst ucto ad Y g s a ucto o dst ucto o Y al h s gv b h g d W ca td s to two-dsoal cas I ad Y ar two cotuous rado varabls w ot dst ucto ad Z φ Y ad W φ Y ar two rado varabls obtad as trasoratos ro Y bvarat dst ucto or Z ad W s gv b z w z φ w φ h dd whr z ad w ar varabls corrsodg to trasoratos φ Y ad φ Y Ths has obvous tso to gral -dsoal cas Khur 4 gav a al o two ddt Gaa rado varabls ad Y so at ad Y ar gaa rado varabls w dstrbutos Γ λ ad Γ λ rsctvl I w dot dsts as ad rsctvl w hav λ λ or > Γ or
6 AAM: Itr J Vol Issu Ju 8 [Prvousl Vol No ] 47 λ λ or > Γ or I at cas w d Z ad W Y Z s dstrbutd as Bta w Y aratrs ad W s dstrbutd as Gaa w aratr valus ad Fro ow o w shall us β a b to dcat a Bta rado varabl w ostv aratrs a b ad Γ c to dcat a Gaa dstrbuto w ostv aratrs c ad Now lt us cosdr rado varabls dstrbutd ddtl so at s dstrbutd as gaa Γ or ad w d Y Y Y Th w hav Y dstrbutd as β Y dstrbutd as β Y dstrbutd as Γ Ths ca b show ollowg actl sa tchqu usd Khur 4 whch s a o st gralzato o rsult rovd b h So w d Th ot dst o Y Y ad Y s gv b Γ Γ Γ g d d d d d d Usg rorts o dlta ucto
7 48 Satau Charabort rost tgral tgral w rsct to d d Nt w us dlta ucto rorts to dal w scod tgral tgral w rsct to as d d Th w us dlta ucto rorts or outrost tgral wout costat trs s d Fall uttg costat trs togr w gt g Γ Γ Γ Ths colts roo 6 Vctor otatos or dlta uctos ultdsoal cas
8 AAM: Itr J Vol Issu Ju 8 [Prvousl Vol No ] 49 I ultdsoal cas trasorato s lar Y A whr Y ad ar ad vctors rsctvl ad A s a atr w ca rss g dst o Y vctor otato trs o dst o as ollows whr So g A d T T A a a T T a whr a ar rows o atr A Now o-dsoal st u a s a scalar a s gv b a Slarl ultdsoal st u a Y A as abov ad A s a osgular atr so at w ust hav A A A Ths s bcaus o ollowg: sc trasorato s osgular w hav g A A ad ror ro A A A d But w ow at A d A Thror ollows Slarl vctor w hav Y A b whr A s a osgular atr ad b s a
9 5 Satau Charabort A b A b A Usg s o ca coclud at s ultvarat oral w μ as a vctor ad as covarac atr or a osgular trasorato A ad a costat vctor b trasord vctor Y A b ollows ultvarat oral w Aμ b as a vctor T ad A A as varac-covarac atr 7 Trasorato o Varabls Dscrt Cas Trasorato o varabls ca b ald to dscrt cas as wll I s a dscrt rado varabl tag valus a a w robablts ad Y g s a trasord varabl corrsodg robablt ass ucto or Y s gv b q g d g a I two-dsoal cas s ot robablt ass ucto or two varabls ad Y q z w ot robablt ass ucto or trasord ar Z φ ad W φ s gv b Y Y z w z φ w φ q dd z φ a b w φ a b Whr ar Y s dscrt havg valus a b w robablts or ad 8 Mots ad Mot Gratg Fuctos I uvarat st u d o-ctral ot o s wrtt as a d a d a I bvarat st u o-ctral ot o ordr l or Y s gv b l l dd a b dd
10 AAM: Itr J Vol Issu Ju 8 [Prvousl Vol No ] 5 l a b dd For al Y ollow troal dstrbuto! l! l! l! l a b l l As a rsult corrsodg o-ctral ot o ordr r s s gv b r s l l l! r s dd a bl! l! l! Th ost trstg art Khur's artcl s rrstato o dst ucto o a cotuous rado varabl trs o ts o-ctral ots Thus s dst ucto or s rrstd as l <! µ whr s gralzd drvatv o ad µ s ordr o-ctral ot or rado varabl O ca s a roo o s rsult Kawal 998 Lt us brl to tchqu usd b h to drv abov rsso H showd at or a ral ucto dd ad drtabl o all ordrs a ghborhood o zro < µ! whr r roduct btw uctos ad s dd as d Ths lads us to coclud 4! µ < Th crucal st roo was to us Talor's aso o-dso or ucto Thus w hav <!
11 5 Satau Charabort Th d Ths two sts gv us rlato 4 Wh w ov to two-dsoal scaro w shall hav to us Talor's srs aso or a two-dsoal aaltc ucto about ot whch s gv b < ] [! C! 5 Now hr also w ollow sa tchqu ad so w cout whch s dd as dd Now usg Talor s srs aso o about ro 5 ad assug at trchag o tgrals w suatos rssbl w gt < dd C ]! [ < dd!! <!! Now w d rs ordr o-ctral ot or ar Y as s r s r s r dd µ Th ro abov <!! µ < µ!!
12 AAM: Itr J Vol Issu Ju 8 [Prvousl Vol No ] 5 Th last qualt ollows bcaus o ollowg rlato Thror w hav < < µ!! Now o-ctral ot o ordr r s s gv b But w also hav r s r s dd < r s dd r or s rs r! s! r ad s Thror o-ctral ot o ordr r s rducs to µ r s µ dd!! Wh w tal about ot gratg ucto o varabl cas w hav t t φ t E µ <! t µ < d d t < µ!! µ t <! d d I two-varabl cas ot gratg ucto s gv b φ s t s ty s t E dd s t < µ <!! s t < µ <!! dd dd
13 54 Satau Charabort µ!! < < < < s s t µ!! t I gral -dsoal cas ot gratg ucto could b obtad actl slar asho so at t s gv b t t t φ t t t µ < < <!!! 9 Cocludg Rars Th stud o gralzd uctos s ow wdl usd ald aatcs ad grg sccs Th -ucto aroach rovds us w a ud aroach tratg dscrt ad cotuous dstrbutos Ths aroach has ottal to acltat w was o ag so classcal cocts aatcal statstcs Howvr so trstg alcatos ca b oud ar b Paza ad Prozato 996 I s ar auors us dlta ucto aroach or dsts o olar statstcs ad or argal dsts olar rgrsso W ar also loog orward to obta so trstg alcatos o dlta ucto statstcs Acowldgt: I a scrl aul to Prossor Loa Dba Uvrst o Tas-Pa Arca or brgg s robl to otc REFERENCES Drac PAM 9 Th Prcls o Quatu Mchacs Oord Uvrst Prss Hoss RF 998 Gralzd Fuctos Ells Horwood Ltd Chchstr Suss Eglad Kawal RP 998 Fucto Thor ad Tchqu d Edto Bosto MA Brhausr Khur AI 4 Alcatos o Drac's dlta ucto statstcs Itratoal Joural o Maatcal Educato Scc ad Tcholog 5 o Paza A ad Prozato L 996 A Drac-ucto od or dsts o olar statstcs ad or argal dsts olar rgrsso Statstcs & Probablt Lttrs Sachv AI ad Woczs WA 997 Dstrbutos Phscal ad Egrg Sccs Bosto MA Brhausr
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