10. Joint Moments and Joint Characteristic Functions

Transcription

1 0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi th rv g 0- Usig 6- w ca dfi th ma of to b µ E z f z dz 0-

2 Howvr th sitatio hr is similar to that i 6-3 ad it is possibl to xprss th ma of g i trms of f x y withot comptig f z To s this rcall from 5-6 ad 7-0 that P z < z z f z z P z < g z z x y D whr D z is th rgio i xy pla satisfyig th abov iqality From 0-3 w gt As z covrs th tir z axis th corrspodig rgios ar oovrlappig ad thy covr th tir xy pla f z x y x y 0-3 zf z z g x y f x y x y 0-4 x y D z D z

3 By itgratig 0-4 w obtai th sfl formla or E zf zdz gxy f xydxdy 0-5 E[ g ] g x y f x y dxdy If ad ar discrt-typ rvs th Sic xpctatio is a liar oprator w also gt 0-6 E[ g ] g xi y j P x i y j 0-7 i j E akgk ake[ gk ] k k 0-8 3

4 If ad ar idpdt rvs it is asy to s that ad W h ar always idpdt of ach othr I that cas sig 0-7 w gt th itrstig rslt E[ g h ] g x h y f g x f x dx x f h y f y dxdy y dy Howvr 0-9 is i gral ot tr if ad ar ot idpdt g I th cas of o radom variabl s 0-6 w dfid th paramtrs ma ad variac to rprst its avrag bhavior How dos o paramtrically rprst similar cross-bhavior btw two radom variabls? Towards this w ca graliz th variac dfiitio giv i 6-6 as show blow: 4 E[ g ] E[ h ] 0-9

5 Covariac: Giv ay two rvs ad dfi Cov E[ µ µ ] 0-0 By xpadig ad simplifyig th right sid of 0-0 w also gt Cov It is asy to s that E µ µ E E E 0- Cov Var Var 0- To s 0- lt so that U a Var U E a [{ } ] a µ µ Var a Cov Var

6 Th right sid of 0-3 rprsts a qadratic i th variabl a that has o distict ral roots Fig 0 Ths th roots ar imagiary or dobl ad hc th discrimiat mst b o-positiv ad that givs 0- Usig 0- w may dfi th ormalizd paramtr or ρ [ Cov ] Var Var Cov Cov ρ Var Var Cov ρ 0-5 VarU 0-4 ad it rprsts th corrlatio cofficit btw ad Fig 0 6 a

7 Ucorrlatd rvs: If ρ 0 th ad ar said to b corrlatd rvs From if ad ar corrlatd th E E E Orthogoality: ad ar said to b orthogoal if E From if ithr or has zro ma th orthogoality implis corrlatdss also ad vic-vrsa Sppos ad ar idpdt rvs Th from 0-9 with g h w gt ad togthr with 0-6 w cocld that th radom variabls ar corrlatd ths jstifyig th origial dfiitio i 0-0 Ths idpdc implis corrlatdss 0 E E E

8 Natrally if two radom variabls ar statistically idpdt th thr caot b ay corrlatio btw thm ρ 0 Howvr th covrs is i gral ot tr As th xt xampl shows radom variabls ca b corrlatd withot big idpdt Exampl 0: Lt U0 U0 Sppos ad ar idpdt Dfi W - Show that ad W ar dpdt bt corrlatd rvs Soltio: z x y w x y givs th oly soltio st to b Morovr ad J z x z w y z w 0 < z < < w < z w z w z > w w / 8

9 Ths s th shadd rgio i Fig 0 f W z w / 0 < z < < w < z w z w w < 0 othrwis w z 0-8 z ad hc Fig 0 f z f W z w dw or by dirct comptatio z z -z z- dw z dw z 0 < z < < z < 9

10 ad f W f z 0 < z < z f z f z z < z < 0 othrwis w fw z w dz dz w < < 0 othrwis w w w Clarly f W z w f z fw w Ths ad W ar ot idpdt Howvr ad ad hc [ ] E E 0 E W E E W E 0 Cov W E W E E W 0 implyig that ad W ar corrlatd radom variabls

11 Exampl 0: Lt a b Dtrmi th variac of i trms of ad ρ Soltio: ad sig 0-5 µ E Ea b aµ bµ Var E µ E a µ b µ ae µ abe µ µ be µ a abρ b 0-3 I particlar if ad ar idpdt th ρ 0 ad 0-3 rdcs to a b 0-4 Ths th variac of th sm of idpdt rvs is th sm of thir variacs a b

12 Momts: rprsts th joit momt of ordr km for ad Followig th o radom variabl cas w ca dfi th joit charactristic fctio btw two radom variabls which will tr ot to b sfl for momt calclatios Joit charactristic fctios: Th joit charactristic fctio btw ad is dfid as Not that k m k m E [ ] x y f x y dx dy 0-5 j v j v v E f x y dxdy 0-6 v 00

13 It is asy to show that If ad ar idpdt rvs th from 0-6 w obtai Also E j Mor o Gassia rvs : v v 0 v 0 j jv v E E v From Lctr 7 ad ar said to b joitly Gassia as N µ µ ρ if thir joit pdf has th form i 7-3 I that cas by dirct sbstittio ad simplificatio w obtai th joit charactristic fctio of two joitly 3 Gassia rvs to b v 0 v 0-9

14 v E j v j µ µ v ρ v v 0-30 Eqatio 0-4 ca b sd to mak varios coclsios Lttig i 0-30 w gt v 0 ad it agrs with jµ From 7-3 by dirct comptatio sig 0- it is asy to show that for two joitly Gassia radom variabls Cov ρ Hc from 0-4 ρ i N µ µ ρ rprsts th actal corrlatio cofficit of th two joitly Gassia rvs i 7-3 Notic that ρ 0 implis 0-3 4

15 f f x f y Ths if ad ar joitly Gassia corrlatdss dos imply idpdc btw th two radom variabls Gassia cas is th oly xcptio whr th two cocpts imply ach othr Exampl 03: Lt ad b joitly Gassia rvs with paramtrs N µ µ ρ Dfi a b Dtrmi f z Soltio: I this cas w ca mak s of charactristic fctio to solv this problm E j E a b j a b E ja jb 0-3 5

16 6 From 0-30 with ad v rplacd by a ad b rspctivly w gt whr Notic that 0-33 has th sam form as 0-3 ad hc w cocld that is also Gassia with ma ad variac as i which also agrs with 0-3 From th prvios xampl w cocld that ay liar combiatio of joitly Gassia rvs grat a Gassia rv j b ab a b a j µ ρ µ µ b a b ab a b a ρ µ µ µ

17 I othr words liarity prsrvs Gassiaity W ca s th charactristic fctio rlatio to cocld a v mor gral rslt Exampl 04: Sppos ad ar joitly Gassia rvs as i th prvios xampl Dfi two liar combiatios a b W c d 0-36 what ca w say abot thir joit distribtio? Soltio: Th charactristic fctio of ad W is giv by W v E E j Wv j a cv E j b dv j a b j c d v a cv b dv 0-37 As bfor sbstittig 0-30 ito 0-37 with ad v rplacd by a cv ad b dv rspctivly w gt 7

18 8 v v v j W W W W v ρ µ µ 0-38 whr ad From 0-38 w cocld that ad W ar also joitly distribtd Gassia rvs with mas variacs ad corrlatio cofficit as i W W d cd c b ab a d c b a ρ ρ µ µ µ µ µ µ W W bd bc ad ac ρ ρ 0-43

19 To smmariz ay two liar combiatios of joitly Gassia radom variabls idpdt or dpdt ar also joitly Gassia rvs Gassia ipt Liar oprator Gassia otpt Fig 03 Of cors w cold hav rachd th sam coclsio by drivig th joit pdf f W z w sig th tchiq dvlopd i sctio 9 rfr 7-9 Gassia radom variabls ar also itrstig bcas of th followig rslt: Ctral Limit Thorm: Sppos ar a st of zro ma idpdt idtically distribtd iid radom 9

20 variabls with som commo distribtio Cosidr thir scald sm Th asymptotically as Proof: Althogh th thorm is tr dr v mor gral coditios w shall prov it hr dr th idpdc assmptio Lt rprst thir commo variac Sic w hav Var N 0 E i i 0 E i

21 Cosidr whr w hav mad s of th idpdc of th rvs Bt whr w hav mad s of Sbstittig 0-49 ito 0-48 w obtai ad as i i j j j E E E i i / / / !! 3/ 3/ / o j j j E E i i i j i / o / lim

22 sic x x lim 0-5 3/ [Not that / trms i 0-50 dcay fastr tha Bt 0-5 rprsts th charactristic fctio of a zro ma ormal rv with variac ad 0-45 follows 3/ o / ] Th ctral limit thorm stats that a larg sm of idpdt radom variabls ach with fiit variac tds to bhav lik a ormal radom variabl Ths th idividal pdfs bcom importat to aalyz th collctiv sm bhavior If w modl th ois phomo as th sm of a larg mbr of idpdt radom variabls g: lctro motio i rsistor compots th this thorm allows s to cocld that ois bhavs lik a Gassia rv

23 It may b rmarkd that th fiit variac assmptio is cssary for th thorm to hold good To prov its importac cosidr th rvs to b Cachy distribtd ad lt whr ach i Th sic sbstittig this ito 0-48 w gt C α i which shows that is still Cachy with paramtr α I othr words ctral limit thorm dos t hold good for a st of Cachy rvs as thir variacs ar dfid 0-53 α 0-54 i α / / ~ C α

24 Joit charactristic fctios ar sfl i dtrmiig th pdf of liar combiatios of rvs For xampl with ad as idpdt Poisso rvs with paramtrs ad λ rspctivly lt 0-56 Th Bt from 6-33 so that i sm of idpdt Poisso rvs is also a Poisso radom variabl j 0-57 λ λ 0-58 j λ λ P λ λ j λ