Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =

Transcription

1 L's rvs codol rol whr h v M s rssd rs o h rdo vrl. L { M } rrr v such h { M } Assu. { } { A M} { A { } } M < { } { } A u { } { } { A} { A} ( A) ( A) { A} A A { A } hs llows us o cosdr h cs wh M { } [ ( A) ( A) ] { A} ( ) ( ) { } l { < < } A A A A l ( ) ( ) A ds Whch s ohg or h h drvv o hus w hv dd { } { A } lhough { } ( A) ( A) A () L us loo or closl hs rsso. W c wr hs s: Now grg oh sds lds: { } A ( A) A ()

2 { } A A d A d - whr h rs grl quls. hs s h couous vrso o h ol rol hor. W c h wr quo () s ( A) { A } { A } B's hor W could lso show s w dd rvousl h wh M { } d A { } Slrl ( ) ( ) ( ) ( ) W hv ohr or o B's hor W wll wr hs s ( ) ( ) d ( ) ( ) d d ( ) ( ) ( ) sc ( ) ( ) ( ) ( ) ( ) ( ) d ol rol: W hv ohr or or ol rol.

3 ro d ( ) ( ) d ( ) d I M { } w could sl d ( ) Codol cd vlus r h wr s; { < < } { < < } ( ) ( ) ( ) ( ) > <.. [ g( ) M ] g ( Md) [ g( ) ] g ( ) g d d d L us cou wh rdo vrls w o l u ow w cosdr wo rdo vrls. l: r rrvs h so h rvl () sog or "" us. r rrvs ddl h s rvl sog or "" us. ) d h rol h wll rrv or. ) d h rol h h rs wll. ) Assug h h d h rol h rrvd or.

4 rs dr h rol sc. A ouco (or l) o h r s h rrvl o rs d scc scs d. S { ll r o urs such h }.. s h v { rrvs h rvl ( )} {} ( ) A A (hs s h g o rdo rrvl) Cosdr ow h v { rrvs h rvl ( 4 )} { } B { B} { } 4 4 ( ) 4 B A 4 A B h roduc B { }{ 4} { } 4 B

5 Assug h ddc o rrvl s { } 4 ( )( ) 4 W s h w hv coll rdo vs h rol o rrr v M s D M { M } whr D M s h r our sl sc. Our r s coll dd. Now!!! h v { rrvs or } { } C loos l h ollowg C { } I's r [ ] Now h v { s } s wh w w o dr. L's dr h v sc. Assu rrvs or -

6 L I h. Ad rrvs rs L - ' I ' or h. { h } { } O our V dgr D D { }

7 D Ar o Ar h v { rrvs or gv h h } W o h [ ] [ ] A CD CD CD Wh h roduco o r.v. w c cosdr grl wo rdo vrls Gv wo rdo vrls d Hc

8 [ ] [ ] h roduc o h wo vs s [ ] [ ] [ ] All oucos ξ such h ( ξ ) d ( ξ ) W c d Jo rol Dsruo uco ( ) { ; } I h sud o svrl rdo vrls h dsruo o ch vrl s usull clld h Mrgl Dsruo I grl s h rgl dsruo o ( ) c o drd ro d Bu s rld o h. ( ) [ ] [ ][ ] [ ] ( ) ( ) ( ). Slrl L us ssu <. { } { } { < } hs sc: { } { < }

9 h ls wo vs r uull clusv. < < < Slrl < < W could lso cosdr h v - W lso hv cd Vlus Mos Chrcrsc ucos s or. I [] [ ] dd g g dz z z z g Z z I dscr [ ] g g

10 [ g ( ) g ( ) g ( ) ] [ g ( ) ] [ g ( ) ] [ g ( ) ] Codol cd Vlus [ g M ] g( ) ( M )dd I r slr o wh w hv do or w could show [ g( ) ] g ( ) ( ) ( ) ( ) [ g( ) ] g( ) ( ) d [ g( ) g ( ) ] g [ g ( ) ] [ g( ) ] [ g( ) ] dd g ( ) ( ) d Mos h o os r o wo rdo vrls r dd r r [ ] r d - ordr o os r ( ) dd η η R - Auo Corrlo Jo Crl Mos M r r dd

11 M M M r r [ ] ( η ) ( η ) ( ) r ( η ) ( η ) M [( η )( η )] [ ] η[ ] η [ ] [ ] [ ] [ ] η η Covrc dd h ro: Γ [( η )( η )] [( η ) ] ( η ) [ ] M - Corrlo Coc I c show r wo rdo vrls r ucorrld [ ] [ ] [ ] orhogol [ ] dd ( ) hor ) I d r dd h h r lso ucorrld. ) I d r ucorrld h hr covrc d corrlo coc r zro. [( η )( η )] r ) I d r ucorrld h h vrc o hr su quls h su o hr vrcs... [( ) ] [ ] [ ] No: I grl g ( ) d ( ) h r wo rdo ucos d d r ucorrld g h g h grl sc s ol grl ror o [ ] [ ] [ ] ( ) Bu h r dd [ g( ) h( )] [ g ] [ h ] ruor vr

12 Mo Grg uco Bor w g w d h ollowg I G ( c) h h [ G ] ( c) o o r.v. roud cos c h I c ( ) o roud org or us rw o I c ( ) h h ( h d o roud h or sl crl o crl o s h vrc) Usg o grg ucos w shll vlu ll rw os wh h s o h r.v. sudd. Do: h o grg uco o r.v. ( ) dd s h cd vlu o h uco wh s sl rrr rl ur. ( ) ( ) ( ) ll d o lws s or ll ohr words c hr or (dos o s). I ollows h rol lw or dsruo o lws ossss o grg uco. Cosqul o d uco clld chrcrsc uco h lws. ss or ll vlus o W wll l lr ou hs or ow w co ourslvs o Mo Grg ucos. o sl ssu M.G.. ss. (No: I M.G.. ss or ll (-L L) h ll os s d h dsruo s uqul drd.)

13 so o!! Hc ( ) ( ) ( ) ( ) ( )! d! Drg wh rsc o Lg ' ( ) ( ) ( ) w g ( ) W dd h M.G.. or r.v. s:! ' rs rw o (h ) Slrl: " ( ) ( ) d ( ) ( ) ( ) " ( ) "' ( ) h ( ) rw o scod rw o hrd rw o rors: ) I hs M.G.. d w r.v. gv whr d r cos h ( ) ( ) roo: B do ( ) [ ] [ ] ( ) QD

14 ) I Z r dd r.v. d r coss h ( ) ( ) ( ) ( ) Z )L d wo r.v. wh M.G.. ( ) d ( ) ( ) rscvl. I or ll vlus o h h dsruos o d r dcl. hs s h or ror. hs s h clog or l o M.G.. c dvlod dg ssocd rol ds ucos. l R.v. s oll dsrud () > ohrws d d vrc usg M.G.. Soluo: h M.G.. s ( ) ( ) ( ) d d hs s h or ol dsruo wh rr h M.G.. ss or <. or rs drvv: ( ) ' d ' ( ) ()

15 or scod drvv: () vrc " " l [ ] d d d [ ] ' hs ds crr ovr o or h o r.v. wo dd r.v. d r chrcrzd s ollows: l lswhr lswhr Drv h dsruo o hr su Z Soluo:

16 No: d d ( ) ( ) ( ) ( ) ( ) s M.G.. o g dsruo wh d r. G Ds uco Γ Γ ( ) () r r Γ () r r ( r ) r! r d r d r oh > Hc ( ) z lswhr ( ) Ovousl: h M.G.. o Z s Z ( ) d W c vlu hs grl lg ( ). L ( ) Z ( ) ( ) ( ) ( ) d d d Γ d ( ) G uco ( )

17 ( ) du d v u dv d ( ) d ( ) ( ) ( ) l ( ) ( ) ( ) ( ) ( ) d L ( ) ( ) ( ) ( ) ( ) ( ) d ( ) u du d dv v ( ) d ( ) ( ) ( ) l ( ) ( ) ( ) ( ) ( ) d ( ) d ( ) [ ] l: Gv d dsruo o. h srs so o ( ) ss or vr vlu o [ ] d

18 Assu h ( ) [ ] d ( ) > l: Suos ( ) ( g ) whr s vlus... d dsruo o. Soluo: so o ( ) s rvl ( ) hc M.G.. drs dsruo. or vr dg ( ) q q ( ) q dg grl rs w hv ( ) ( ) q q l: Gv orll dsrud r.v. wh π d M.G.. ( ) s ( ) µ µ Assug s dd r.v. ch wh µ d vrc

19 d ).d.. o ).d.. o B do d Bu d µ ϑ µ ϑ ϑ µ µ ϑ Bu µ µ µ µ hs s chrcrsc o h Norl Dsruo s orl µ µ π Wh µ ϑ µ ϑ µ ϑ µ

20 Norl µ µ π l: vlu h Mo Grg uco or h gorc dsruo d us o d h d vrc o h dsruo. ( ) () () M ( ) ( ) ( ) ( ) Sc su s gorc rogrsso So ' ' () [ ( ) ] ( ) ( ) [ ( ) ] [ ] [ ] [ ( ) ]

21 or () [ ] [ ] [ ] [ ] [ ] 4 4 " " V l: I vlus d wh q q l: Bol q q q l: I osso!! L us roduc Chrcrsc ucos h Chrcrsc uco o rdo vrl s h ourr rsor o s ds uco (wh rvrsl sg) I s usd o sl hcl oros volvg h rdo vrl. vluo o os. Dro o dss. Covoluo w wo dss

22 Dd ( ) [ ] Couous Dscr - cd vlu o cos s d [ ] I s rqul cov o us h url logrh o ( ) β ( ) l ( ) Scod Chrcrsc uco W o h ( ) d d sc ( ) ( ) d d or l ( ) ( ) [ ] d ( ) ( ) [ ] d d d

23 Ivrso orul h ds c rssd rs o ( ) ( ) d π h Dsruo uco c rssd rs o ( ) ( ) d ( ) π π ( ) d dd I - v h ( ) s rl d v ( ) π ( ) cosd cosd h coc o chrcrsc uco c usd o dr h ds o g whou vlug ( ) or ( ). or l Bu ( ) g ( ) [ ] d (*) d Wh chg o vrls (*) d susuo g() o (*) w c wr s h d. h sc uco s uqul drd rsor

24 hs hod hs h dcul h h rsor g() s o lws o or o. Hc og h or d h s o lws sl od. Howvr h hod so s lds o sl rsuls s h ollowg l shows. l: Assu π d d d d d d d g d > π π d s o or o sc gos ro our rvl. U h h d U d d π π π π

25 Mo hor d ol!!! whr d W c s h w h drvv o h ov quo wh rsc o s w g h ollowg: d d....! d d d d d d d d d d d d W s h w ow h chrcrscs uco w c dr h os o h r.v. l: I hs ol dsruo q

26 ' " ' " ( ) ( q) [ q] ( ) ( )( q) [ q] ( ) ( q) [ q] [ ] ( ) [ ] [ ] [ ( ) ] [ q] [ ] q q Covoluo I w r gv wo ucos d o wr w c or h grl ( ξ ) ( ξ ) dξ ( ξ ) ( ξ ) dξ d s ow s h covoluo o d I suggs ou rr o our h or dl dscusso W o so sc rors: < d < or < () h () Slrl > c d > d h or > c d I ( ) s chrcrsc uco o ( ) ( ) s chrcrsc uco o Covoluo hor h ( ) ( ) ( ) chrcrsc uco o hr covoluo

27 roo d ( ) ( ) dd dd L τ τ ( τ ) τ ( τ ) ( τ ) ( ) ( ) τ dτd dτd dτd QD rol Grg ucos A or clss o dscr rdo vrls s o gr vlus hs rrs "cou" o sohg wg ls ucurg hosl srvc oulo growh. A hcl dvc usul dg h rol dsruos d ohr rors o gr vlud r.v. s h rol Grg uco dd s () [ ] L gr vlud rdo vrl wh h ( ) () [ ] ) (dd rs o ro. [ ]

28 I () s ow d w c d o srs h w c dr () s h cocs o. Rd dro wh rsc o ld corl os or r.v. [ ( )( ) ( ) ] µ s osv gr Wh rol Grg uco ss c drd roud () ( ) () () () ( ) () µ Slrl I grl ( ) () ( ) ( ) () ( ( ) ) µ [ ] ( ) () ( )( ) ( ) l d.g.. or gorc r.v. d d sc c' o s vlu h () [ ] q ( q) q q [ q ( q) ( q) ] gorc rogrsso W c l (rrr) so h q < q q [ q{ q ( q) } ] q

29 Now () () () () s d d q q ( q) ( q) ( q) ( q) [ ] () rovds ohr ool or dg os o rdo vrls. I's ggs ss s h c usd o drv rol ucos whch w s lr o. Sg w words ou Codol Dsruos d Dss W rcll h h ( ) { } ( ) ( ) W d; - Codol dsruo ( ) whr { } o h rdo vrl ( ) { } { } ( ) s h v o ll oucos ξ such h ( ξ ) ξ d... h s roduc o h vs { } - All ohr rors s. ( ) ( ) { } { } { } - h ssoc ds uco s dd ( ) d ( ) d d hs ll h rors o ordr ds uco.

30 l: r d r h rdo vrl s dd ( ) 6 wh { } { v ur} 4 6 W wll d { } < 4 <. h { } { 4 6} { 4 6} { 4 6} { } { 4} { 4} { } { } { } 6 { } 6 { } { } 6 6 () () W c lso l ou ol codol rol. W hd B B B whr S d h r uull clusv. (A ro o h sc S)

31 I w l B L us ow cosdr h v rssd rs o h rdo vrl. L Assug [ ] [ ] or h lso lss s h lss s sc h ξ I < [ ] < h us lss h lss s sc ξ () Corrsodg ds uco < d

32 W l dr d o uco o : < whr W c ow show slrl [ ] < < < < < < d () < () hs lls us h sc w ow ddol oro ou w r or cr ou g ss sd rg o. l s orll dsrud d η Sc ) ( η η η η η

33 w hv { η } ( η ) π ( ( ) ) lswhr or η < η π π () ( η ) () η η η