Extension of Two-Dimensional Discrete Random Variables Conditional Distribution

Transcription

1 Itratoal Busss Rsarch wwwccstorg/br Extso of Two-Dsoal Dscrt Rado Varabls Codtoal Dstrbuto Fxu Huag Dpartt of Ecoocs, Dala Uvrsty of Tchology Dala 604, Cha E-al: Chg L Dpartt of Ecoocs, Dala Uvrsty of Tchology Dala 604, Cha Ths wor s supportd by th Natoal Scc Foudato udr Grat No ;th fuds proct udr th Mstry of Educato of th PRC for youg popl who ar dvotd to th rsarchs of huats ad socal sccs udr Grat No 09YJC79005;softwar + rsarch of Dala Uvrsty of Tchology udr Grat No 8430 Abstract Codtoal dstrbuto rflcts th dpdcy l aog rado varabls, but two-dsoal rado varabls Codtoal Dstrbuto has so ltatos I ordr to rch th cott of codtoal dstrbuto ths papr gvs th xtso of codtoal dstrbuto udr dscrt rado varabls ad so xapls Ths artcl obtas th xtso strctly accordac wth th dfto of two-dsoal rado varabls So t ca gt codtoal dstrbutos aftr chagg th codto ad gt codtoal dstrbutos that ar xtdd to -dsoal rado varabls, thrby rchg th cotts of th codtoal dstrbuto Kywords: Dscrt Rado Varabls, Two-dsoal Rado Varabls, Codtoal Dstrbuto Itroducto Two-dsoal rado varabls codtoal dstrbuto s th dstrbuto of o varabl wh aothr varabl s a fxd valu Th rlatoshps of two-dsoal rado varabls (, Y) ar aly dvdd to two typs: dpdc ad dpdc Th or valuatos ad Y hav, th or codtoal dstrbutos wll b thr Each codtoal dstrbuto dscrbs o stat-spcfc dstrbuto fro a sd So th cotts of codtoal dstrbuto ar rchr ad ts applcatos ar broadr For ay ssus valus of cocrd rado varabls td to fluc ach othr, ths as codtoal dstrbuto as a powrful tool for studyg dpdcs aog varabls Codtoal dstrbuto for rado varabls drvs fro codtoal probablty for rado vts, so thr s a clos rlatoshp btw th two ad th approachs to hadl th ar th sa, but codtoal dstrbuto s or coplx to dal wth I rct yars, th a rsarch drcto Cha s th rsarch of codtoal gvalus ad th xtso ad applcato of codtoal dstrbuto Codtoal gvalus ar aly potd to codtoal xpctato, whch s xpctato udr codtoal dstrbuto I ths ara th a cotrbutos ar Wag Chg, Zou Hal gav th dfto ad rsarchd th charactrstcs of rado varabls codtoal partcular fucto basd o asurt ad tgral thory; Zhag M usd u a-squard rror to solv o d probls of bst prdcto ad too xapls to aalyz th applcato of codtoal xpctato practcal prdcto probls; u hu ad Wu Guogg ducd total probablty forula of dscrt ad cotuous rado varabls basd o codtoal xpctato ad dcatv fucto I A of rado vts A For xtso of codtoal dstrbuto, pracy rsarchs ar Chg Fyu gralzd th codto dstrbuto of th posso procss arrval t; Yag Jgpg, tc vstgatd th argal rcursv quatos o xcss-of-loss rsurac traty udr th assupto that th ubr of clas blogs to th faly cosstg of Posso, boal ad gatv boal, ad that th svrty dstrbuto had boudd cotuous dsty fucto; Hu Duapg gav a xprss forula of dstrbuto for llptcally cotourd atrx dstrbutos ad showd that codto dstrbuto of llptcally cotourd atrx dstrbuto s llptcally cotourd dstrbuto yt I th abov study, thy focusd o th applcato of th codtoal 60

2 Itratoal Busss Rsarch Vol 3, No ; Aprl 00 dstrbuto showg th rsarch sgfcac of th codtoal dstrbuto, but thr s fw studs rsarchg th atur of th codtoal dstrbuto I coparso, th studs of codtoal dstrbuto abroad ar or -dpth For xapl, Rody CL Wolff, tc studd th thods of valuatg codtoal dstrbuto fucto; Jusha Ba vstgatd th dyac odl of tstg paratrc codtoal dstrbutos Ptr Hall ad Qw Yao dscussd approxatg codtoal dstrbuto fucto usg dso rducto Bruc E Has studd oparatrc stato of sooth codtoal dstrbutos [] ; Prs Dacos ad Brd Sturfls aalyzd codtoal dstrbutos usg algbrac algorths for saplg Thy hav show that th rsarchs of codtoal dstrbuto ar ult-factd ad or coplx whl a agast udrgraduat tachg Thrfor, ths papr bgs to dscuss ad aalyz fro th basc cott of codtoal dstrbuto ad ducs gral forulas wth crta codtos o th bass of th dfto of codtoal dstrbuto It frst starts for th two-dsoal rado varabl codtoal dstrbuto ad chags th gv codtos to obta th xtsos of codtoal dstrbuto ad th gvs xtsos of codtoal dstrbuto wh thr ar thr-dsoal rado varabls Ths papr s to solv th codtoal dstrbuto of ultdsoal rado varabls udr th gv codtos ad ts rsults ca b usd for tachg, xpdg th owldg of th codtoal dstrbuto ad facltatg popl s calculatos Extso of dscrt rado varabls codtoal dstrbuto Extso St ad Y for th dscrt rado varabls ad, Y ar dpdt Kow th dstrbuto srs of ad Y, udr th gv codto of + Y th codtoal dstrbuto of s P(, + Y ) P( ) P( Y ) P( + Y ) () P + Y P + Y Exapl I) basal xapl: I th cas of two-dsoal rado varabls, ad Y ar dpdt Y P Gv th codto of + Y to solv th codtoal dstrbuto of ad P( ), ( ) To solv: For th su of th dpdt Posso varabls s stll Posso varabl, vz Y P( ) (, + ) P( + Y ) P Y P + Y ( ) ( ) P( + Y ) P P Y, 0,,, + + ( ) ( + ) ( ) ( ) ( + ) + That s udr th codto of Y p + II) Two-dsoal dscrt rado varabls (, ) Gv th codto of To solv: (, ) + +, so +, subcts to boal dstrbuto b(, p ) to solv th codtoal dstrbuto of Y (), thrto Y subct to troal dstrbuto M ( p,, p, p 3) Y subct to troal dstrbuto M ( p,, p, p 3), th thrs ot dstrbuto s P(, Y ) p p ( p p),,,,,, + (3) ( ) For th argal dstrbuto of ultoal dstrbuto s stll ultoal dstrbuto ad th argal, Y b, p dstrbuto of troal dstrbuto s boal, so b( p ), 6

3 Itratoal Busss Rsarch wwwccstorg/br (, ) P( x ) P x y P y x ( ) ( ) p p p p p p p p ( ) p ( p) ( ) ( ) p p p (4) p p That s udr th codto of, Y subcts to boal dstrbuto b(, p), thrto p p p Extso St YZ,, for th dscrt rado varabls ad YZ,, ar utual dpdt Kow th dstrbuto srs of YZ,,, udr th gv codto of + Y + Z th codtoal dstrbuto of s (, + + ) P( + Y + Z ) P Y Z P + Y + Z Exapl I),, ( ) ( + ) P( + Y + Z ) P P Y Z (5) YZ ar utual dpdt, ad P( ), Y P( ), ) Gv th codto of Y + Z to solv th codtoal dstrbuto of Z P To solv: For th su of th dpdt Posso varabls s stll Posso varabl, vz Y Z P( ) P(, Y + Z ) PY ( + Z ) That s stll subcts to P ( ) ( ) P Y + Z P P (6) 3 + +, so ) Gv th codto of + Y + Z to solv th codtoal dstrbuto of To solv: For th su of th dpdt Posso varabls s stll Posso varabl, + Y + Z P + +, so vz ( 3) P(, + Y + Z ) P( + Y + Z ) P( + Y + Z ) That s udr th codto of Y Z p ( ) ( + ) P( + Y + Z ) P P Y Z ( ) + 3 ( + 3) ( ) ( + + 3) ( ) ( + + ) ( + 3) , 0,,, , subcts to boal dstrbuto b(, p ) 3 (7), thrto + +, Y subcts to boal dstrbuto (, ) b p, I a slar way, udr th codto of Y Z thrto p ; Z subcts to boal dstrbuto b(, p 3 ), thrto 3 p II),,, ar utual dpdt ad P( ),,,, Gv th codto of + + to solv th codtoal dstrbuto of 6

4 Itratoal Busss Rsarch Vol 3, No ; Aprl 00 P, To solv: P P P( ) P ( ) P ( ), 0,,, That s udr th codto of + +, p (8) subcts to boal dstrbuto b( p ),, thrto Wh, p p p, b,,,,, Th soluto of th abov two xtsos s rlatvly spl so t s ottd 3 Coclusos Ths papr aly dscusss codtoal dstrbutos of ultdsoal rado varabls ad ts rlatd xapls gv crta codtos th cas of dscrt stuato It chags th orgal codto of o fxd varabl to or coplx codtos, for xapl th codto that th su of two varabls s fxd two-dsoal stuato I addto, codto of dscrt rado varabls ths papr xtds two-dsoal to thr ad -dsoal rado varabls ad gvs th codtoal dstrbutos For xapl, th cas of thr-dso, w ca gt o varabl s codtoal dstrbuto gv th su of th othr two or th thr varabls fxd It s th sa ult-dso Ths artcl gts th abov rsults strctly accordg to two-dsoal rado varabls codtoal dstrbuto Codtoal dstrbuto ca b appld th lf ad wor to rsolv practcal probls Th applcato of usg codtoal dstrbuto thory to carry out sctfc aalyss ad calculatos wth ral data s a portat rflcto of th usfulss of codtoal dstrbuto I addto, for thr ar or ad or dscussos about ultdsoal rado varabls ralty ths papr xtds th codtoal dstrbuto to provd w tra of thought for th rsarch so xtt ad t as at rchg th cott of codtoal dstrbuto, dpg th udrstadg of t ad applyg t wll practcal Rfrcs Bruc E Has (009) Noparatrc Estato of Sooth Codtoal Dstrbutos [EB/OL] (004-05)[7-30] Ch, F-yu (006) Gralzato of th Codto Dstrbuto of th Posso Procss Arrval T Joural of Chag Sha Uvrsty of Elctrc Powr (Natural Scc), ():75-79 Hu, Dua-pg (00) Th Codto Dstrbuto of Ellptcally Cotourd Matrx Dstrbutos Coucato o Appld Mathatcs ad Coputato, 5():

5 Itratoal Busss Rsarch wwwccstorg/br Jusha Ba (003) Tstg Paratrc Codtoal Dstrbutos of Dyac Modls Th Rvw of Ecoocs ad Statstcs, 85(3): Lag, Y (998) Codtoal Probablty ad Codtoal Dstrbuto Joural of Hghr Corrspodc Educato (Natural Sccs),(5):9-3,5 Mao, Sh-sog, Chg, Y-g & Pu, ao-log (004) Probablty Thory ad Mathatcal Statstcs Tutoral Bg: Advacd Educato Prss Prs Dacos & Brd Sturfls (998) Algbrac algorths for saplg fro codtoal dstrbutos Th Aals of statstcs, 6(): Ptr Hall & Qw Yao (005) Approxatg Codtoal Dstrbuto Fuctos Usg Dso Rducto Th Aals of statstcs, 33(3):404-4 Rody CL Wolff, Qw Yao & Ptr Hall (999) Mthods for Estatg a Codtoal Dstrbuto Fucto Joural of th Arca Statstcal Assocato, 94(445):54-63 Wag, Chg & ou, Ha-l (008) So Proprts of Codtoal Charactrstc fucto Joural of Cha Jlag Uvrsty, 9(4): u, Hu & Wu, Guo-g (004) Full Probablty Forula Basd o th Codtoal Expctato ad ts Appllcatos Joural of East Cha Isttut of Tchology, 7():93-95 Yag, Jg-pg, Wag, ao-qa & Chg, Sh-hog (006) Codtoal Rcursv Equatos o Excss-of-Loss Rsurac Appld Mathatcs ad Mchacs, 7(8): Zhag, M (006) Usg Codtoal Expctato to solv Probls of Bst Prdcto Joural of Shaax Isttut of Educato, ():