Convolution of Generated Random Variable from. Exponential Distribution with Stabilizer Constant

Transcription

1 Appld Mamacal Scc Vol 9 5 o HIKARI Ld wwwm-acom p://dxdoog/988/am5559 Covoluo of Gad Radom Vaabl fom Expoal Dbuo w Sablz Coa Dod Dvao Maa Lufaa Oaa ad Maa Aa Dpam of Mamac Facul of Mamac ad Naual Scc Adala Uv Lmau Ma Campu Padag Idoa 5 Copg 5 Dod Dvao Maa Lufaa Oaa ad Maa Aa T acl dbud ud Cav Commo Abuo Lc wc pm ucd u dbuo ad poduco a mdum povdd ogal wo popl cd Abac T w dbuo coucd b gag adom vaabl fom a xpoal dbuo w ablz coa I pd covoluo fom dcall ad dpd w adom vaabl b ug aalcal mod fom of pobabl d fuco ad cumulav dbuo fuco I addo pd om pop of xpoal dbuo w ablz coa ad covoluo cludg ma vaac mom gag fuco caacc fuco w ad uo Mamac Subjc Clafcao: 97K5 E5 A5 Kwod: covoluo xpoal dbuo ablz coa Ioduco T xpoal dbuo a couou pobabl dbuo w mmol pop ad maud falu a I ma dbuo wdl ud poblm of labl uc a o ow oluo of ququ w vc m a um of xpoal adom vaabl T dbuo of um of dpd adom vaabl calld covoluo ad acal mod o oba dbuo av b oducd Fll [5] b ug dfo o pop of mom gag fuco T dbuo of covo-

2 78 Dod Dvao al luo fom xpoal adom vaabl a b dcud b Aouc [] w pobabl dbuo fuco obad b ug mom gag fuco appoac w duco ad aalcal mod Tfo covoluo of xpoal dbuo w am paam obvoul obad b ug caacc fuco ad pop of f dvbl bcomg gamma dbuo T ac o covoluo of xpoal dbuo ad pop a aacd ma ac bcau of applcao o fld uc a Jaulwcz ad Kodc [] av ow covoluo of Elag ad Pacal dbuo a a xpoal dbuo faml labl Mawl T aalcal cm o oba covoluo of xpoal dbuo a ow b Ma ad Lu [] wl Smal al [] oducd poxpoal dbuo a covoluo of xpoal dbuo w dc paam b ug Laplac afom pop T covoluo lad o xpoal dbuo av oducd b pvou pap dvlopd bad o abld dbuo g o ow covoluo ao wa uc a b coucg w pobabl dbuo of gad adom vaabl I acl w dvd covoluo fom gag adom vaabl fom a xpoal dbuo w ablz coa T poc of couco w of pobabl d fuco ad pop xpod Sco ad fumo covoluo ad pop Sco Gad Radom Vaabl fom Expoal Dbuo L X b adom vaabl fom a xpoal dbuo w paam λ Pobabl d fuco of adom vaabl gv b f x fo pov λ ad all pov al umb x do a adom vaabl lad o x wc wa gad fom xpoal dbuo w couou valu pad fom x w x w dfd xm max { x x x} fo x xm] ad m Z T adom vaabl a pobabl d fuco ad cumulav dbuo fuco pcvl a follow m f w x w F w w

3 Covoluo of gad adom vaabl 78 fo pov λ ad w w θ a ablz coa o maa au of pobabl d fuco T ablz coa w oducd dbuo obad a T pop of dbuo a gv xpo of ma vaac mom gag fuco ad caacc fuco pcvl a follow M fo fo T mo fomav pop o xp ap of dbuo a w ad uo pcvl followg fom p p p Sw 5 Ku w fuco fom of w ad uo a dfd a follow p p p

4 78 Dod Dvao al T ap of xpoal dbuo w ablz coa a ow Fgu fo pobabl dbuo fuco ad Fgu fo cumulav dbuo fuco wc vaou paam λ T w dbuo w uppo ] a o ap paam c ol a o ap ca b ougou fomula of w ad uo dpd ol o λ T gap of pobabl dbuo fuco alwa covx a cd o g ax o x a λ dca valu w cal paam /λ Fgu : Pobabl d fuco of adom vaabl wc vaou paam λ Fgu : Cumulav dbuo fuco of adom vaabl wc vaou paam λ Pobabl D Fuco fo Covoluo of Gad Radom Vaabl fom Expoal Dbuo T am of co o p pobabl d fuco ad pop fo covoluo of gag adom vaabl fom a xpoal dbuo w ablz coa Tom L b dpd ad dcall xpoal dbuo w ablz coa w pobabl d fuco dfd a follow f w fo pov λ w ad T um of adom vaabl S a pobabl d fuco f fo! w Poof T om povd b ug aalcal dc mod ad pop of xpcao a b ug dfo of jo dbuo of adom vaabl fom dpd ad dcall xpoal dbuo w ablz coa a

5 Covoluo of gad adom vaabl 785 f w w w w w w L couou fuco o al l ad w df a d a follow I E w w w L w fo ad old I w w g w w w w dw dw dw d d d w dfd fuco g o mplf oao Now w mg cag ax of caa cooda - o pcal cooda b followg fom co co co co w a pcal adu a lcd fo Fumo l u df fo ad uc a obad co ad fo T w av Jacoba fom cagg vaabl poc a follow J ad alo obad co av dfd S a covoluo of adom vaabl m of S w w uppo of adom vaabl S ] Sc w av S w ca w d I w bouda gal fo ] a follow

6 78 Dod Dvao al! co d d d I T poc of cagg vaabl S lad u o w d I folowg fom!! d g d I T d gv xpcao of adom vaabl S w pobabl d fuco f! fo Coolla L b dpd ad dcall xpoal dbuo w ablz coa w pobabl d fuco dfd a follow w w f fo pov λ w ad T um of adom vaabl S fo w pobabl d fuco f! a cumulav dbuo fuco! F Poof T cumulav dbuo fuco dvd fo w av! d F L w u cag vaabl w oba

7 Covoluo of gad adom vaabl 787!! d F w Γ λs low compl gamma fuco T pop of covoluo dbuo fom gad adom vaabl fom a xpoal dbuo w ablz coa a gv xpo of ma vaac mom gag fuco ad caacc fuco pcvl a follow! S!! S! M S fo! S fo T mo fomav pop o xp ap of covoluo dbuo a w ad uo pcvl followg fom! c c c Sw S! c c c c S Ku w fuco fom of w ad uo a dfd a follow c! c! c c! c! c! c

8 788 Dod Dvao al T ap of covoluo dbuo of gag adom vaabl fom a xpoal dbuo w ablz coa av ow Fgu fo pobabl d fuco wc vaou paam λ ad Fgu 5 wc vaou -fold covoluo Fgu fo cumulav dbuo fuco wc vaou paam λ ad Fgu wc vaou -fold covoluo T dbuo w uppo ] w a umb of covoluo a ap paam ad cal paam /λ T ap ad cal paam ca b ougou fomula of w ad uo dpd o ad λ Fgu : Pobabl d fuco of adom vaabl S wc vaou paam λ Fgu : Cumulav dbuo of adom vaabl S wc vaou paam λ Fgu 5: Pobabl d fuco of adom vaabl S wc vaou -fold covoluo Fgu : Cumulav dbuo of adom vaabl S wc vaou -fold covoluo Acowldgm T auo would l o a f ad do fo lpful uggo wc mpovd qual of pap Rfc [] H Jaulwcz ad Kodc Covoluo of lag ad of pacal dbuo w applcao o labl Dmoao Mamaca -8

9 Covoluo of gad adom vaabl 789 [] K Smal T Kad ad S Kad Hpoxpoal dbuo w dff paam Appld mamac - p://dxdoog//am87 [] M Aouc O covoluo of xpoal dbuo Joual of Cugcog Mamacal Soc Vol No [] N-Y Ma ad F Lu A ovl aalcal cm o compu -fold covoluo of xpoal-um dbuo fuco Appld Mamac ad Compuao p://dxdoog//jamc887 [5] Fll A Ioduco o Pobabl To ad I Applcao l Nw Yo 97 Rcvd: Ju 5 5 Publd: Jul 8 5