Probability Theory 2 Lecture Notes - pi.math.cornell.edu

1 Ersonaleducationaluseonlyandarenottob ,Markovchains,andergo dictheorycomesdirectlyfromthecoursetext, o okMarkovChainsandMixingTimesbyDavidLevin ,YuvalPeres,andElizab ' :ThroughProp :Throughde :FinishedSection2 Day8 (Skipp edPolya'sUrn)Day9 :Through rstobservationsab outuniformintegrabilityDay11 :FinishedSection5 Day14 :FinishedSection7 Day17 (Lionel)Day18 (Dan)Day19 :StudentPresentationsDay25:Throughcut-o phenomenonDay26 :FinishedSection11(Skipp edSections12and13)Day28 :ThroughB(d) = i=0Fi(d)ford :FinishedSection15 Day32 :ThroughProp (andsomediscussionofremainingmaterial)Da ys37-42 ( ,F,P)b eaprobabilityspaceandsupp osethatA,B FwithP(B)> ,welearnthattheprobabilityofAconditional onBisde nedasP(A|B) =P(A B)P(B).

2 TheideaisthatifwelearnthatBhaso ccurred,thentheprobabilityspacemustb eup ,thesamplespaceb ecomesB,the -algebranowincludesonlythoseeventscontai nedinB,FB={E B:E F}, (P( |B)alsode nesaprobabilityon( ,F),anditisoftenmoreconvenienttoadoptthi sp ersp ective.)Whenthinkingab outconditionalprobability,itcanb ( ,F,P)asdescribingarandomsystemwhosechanc eofb einginstate ossibleconclusionsthatcanb edrawnab outthestateofthesystem:Allthatcanb esaidiswhetheritliesinAforeachA osethattheobserverhasp erformedameasurementthattellsherifBholds forsomeB FwithP(B) (0,1).IfshefoundoutthatBistrue,herassess mentoftheprobabilityofA Fwouldb eP(A|B).

3 IfshefoundthatBisfalse,shewouldevaluatet heprobabilityofAasP(A|BC).Thus,fromourp ointofview,herdescriptionoftheprobabilit yofAisgivenbytherandomvariableXA( ) ={P(A|B), BP(A|BC), / nitionofP(A|B)totheevents{X=x}and{Y=y}fo rdiscreterandomvariablesX,Yinordertode netheconditionalmassfunctionofXgiventhat Y=yaspX(x|Y=y) =pX,Y(x,y)pY(y).Onethenextrap olatestoabsolutelycontinuousXandYbyrepla cingmassfunctionswithdensities(whichispr oblematicinthatittreatsp dfsasprobabilitiesandraisesissuesconcern ingconditioningonnullevents).Finally,con ditionalexp ectationisde nedintermsofintegratingagainsttheconditi onalpmfs/p ,wewillneedamoresophisticatedtheoryofcon ditioningthatavoidssomeofthepitfalls,par adoxes,andlimitationsoftheframeworksketc hedoutab erde nitionbywayofmorefamiliarconcepts,wewill b eginwithaformalde nitionandthenworkthroughavarietyofexampl esandrelatedresultsinordertoprovidemotiv ation,buildintuition, ( ,F,P)b eaprobabilityspace,X.}

4 ( ,F) (R,B)arandomvariablewithE|X|< ,andG Fasub- neE[X|G],theconditionalexpectationofXgiv enG,tob eanyrandomvariableYsatisfying(i)Y G( ecttoG)(ii) AY dP= AXdPforallA GIfYsatis es(i)and(ii),wesaythatYisaversionofE[X|G ].3 Ourmostimmediateorderofbusinessistoshowt hatthisde nitionmakesgo o ,we rsttakeamomenttoestablishintegrabilityfo rrandomvariableswhich tthede esconditions(i)and(ii)inthede nitionofE[X|G], {Y 0} G,condition(ii)implies AY dP= AXdP A|X|dP, AC( Y)dP= ACY dP= ACX= AC( X)dP AC|X| |Y|= AY dP+ AC( Y)dP A|X|dP+ AC|X|dP=E|X|< . Ournextresultmakesuseofafamoustheoremfro manalysiswhosepro ofcanb (Radon-Niko dym).

5 If and are - nitemeasureson(S,S)with ,thenthereisameasurablefunctionf:S Rsuchthat (A) = Afd foral lA ledtheRadon-Nikodymderivativeof withrespectto ,writtenf=d d .Thefollowingexistencepro ofgivesaninterpretationofconditionalexp ectationintermsofRadon-Niko ( ,F,P)beaprobabilityspace,X: ( ,F) (R,B)arandomvariablewithE|X|< ,andG Fasub- (i)Y G(ii) AY dP= AXdPforal lA osethatX ne (A) = AXdPforA |Gand are nitemeasureson( ,G).(That iscountablyadditiveisaneasyapplicationof theDCT.)Moreover, isclearlyabsolutelycontinuouswithresp dymtheoremthereforeimpliesthatthereisafu nctiond dP Gsuchthat AXdP= (A) = Ad dPisaversionofE[X|G].

6 ForgeneralX,writeX=X+ X andletY1=E[X+|G],Y2=E[X |G].ThenY=Y1 Y2isintegrableandG-measurable,soforallA G, AY dP= AY1dP AY2dP= AX+dP AX dP= AXdP. *Thepro dymtheoremholdsforsignedmeasuresand (A) = Agd de nesasignedmeasurefor nitionofconditionalexp ectation, osethatY isalsoaversionofE[X|G].Condition(ii)impl iesthat AY dP= AXdP= AY dPforallA (i),theeventA ={Y Y }isinGforall >0,hence0 = A Y dP A Y dP= A (Y Y )dP P(Y Y ).ItfollowsthatY Y intheprecedingargumentshowsthatY ,andthepro ofiscomplete. [X|G]andY GwithY=Y ,thenY isalsoaversionofE[X|G]. areG-measurable,E={ :Y( )6=Y ( )} (E) = 0,weseethatforanyB G, BXdP= BY dP= B EY dP+ B\EY dP= B\EY dP= B\EY dP= B\EY dP+ B EY dP= BY dP.

7 , ,andProp ectationisuniqueasanelementofL1( ,G,P).JustaselementsofLpspacesarereallye quivalenceclassesoffunctions(ratherthans p eci cfunctions)inclassicalanalysis,condition alexp eci almostsure quali cationwhensp eakingofrelationsb etweenconditionalexp ecta-tions,butitisimp ,wecanoftenworkwithconvenientversionsofE [X|G]whenweneedtomakeuseofp ,sub- -algebrasrepresent(p otentiallyavailable)information-foreachA GwecanaskwhetherornotAhaso ersp ective,wecanthinkofE[X|G]asgivingthe b estguess G,thenourheuristicsuggeststhatE[X|G] =XsinceifweknowX,thenourb esthede nitionasXalwayssatis escondition(ii)andcondition(i)

8 Ecttoany -algebra,takingX=cshowsthatE[c|G] = ,supp osethatXisindep endentofG-thatis,forallA G,B B,{X B}andAareindep ,Gtellsusnothingab outX,soourb estguessisE[X].Asaconstant,E[X]automatic allysatis escondition(i).Toseethat(ii)holdsaswell, notethatforanyA G, AE[X]dP=E[X]P(A) =E[X]E[1A] =E[X1A] = AXdPbyindep ,ordinaryexp ectationcorresp ondstoconditionalexp { , }. onourintro ductoryexample:Supp osethat 1, 2,..isacountablepartitionof intodisjointmeasurablesets,eachhavingp ositiveprobability( ).LetG= ( 1, 2,..).WeclaimthatE[X|G] =P( i) 1E[X; i]on icontainstheoutcome,andgiventhatinformat ion,ourb estguessforXisitsaverageover ,notethatE[X|G]( ) = iE[X; i]P( i)1 i( )isG-measurablesinceeach ib ,sinceeachA Gisacountabledisjointunionofthe is,itsu cestocheckcondition(ii) iP( i) 1E[X; i]dP=E[X; i] = nitionP(A|H) =E[1A|H],thentheab ovesaysthatP(A|G) =P( i) 1 i1 AdP=P(A i)P( i)on eseenasasp ecialcaseofourde nitionbytakingE[X|Y] =E[X| (Y) ].

9 Toseehowthiscompareswiththede nitiongiveninundergraduateprobability,su pp osethatXandYarediscretewithjointpmfpX,Ya ndmarginalspX, (Y)isgeneratedbythecountablepartition{Y= y}y Range(Y),sothepreviousexampleshowsthatif E|X|< ,thenE[X|Y] =P(Y=y) 1E[X;{Y=y}] =1P(Y=y) xxP(X=x,Y=y)= xxpX,Y(x,y)pY(y)on{Y=y}. ,supp osethatXandYarejointlyabsolutelycontinuouswithjointdensityfX,YandmarginalsfX, oseforsimplicitythatfY(y)>0forally ,ifE|g(X)|< ,thenE[g(X)|Y] =h(Y)whereh(y) = g(x)fX,Y(x,y)fY(y) ob-DynkinlemmashowsthatE[g(X)|Y] (Y).Toseethatthesecondcriterionissatis edaswell,recallthateveryA (Y)isoftheformA={Y B}forsomeB {Y B}h(Y)dP= Bh(y)fY(y)dy= 1B(y)( g(x)fX,Y(x,y)fY(y)dx)fY(y)dy= g(x)1B(y)fX,Y(x,y)dxdy=E[g(X)1B(Y)] = {Y B}g(X) >0isactuallyunnecessarysincetheab ovepro ofonlyneedshtosatisfyh(y)fY(y) = g(x)fX,Y(x,y)dx,sohcantakeonanyvalueatth oseywithfY(y) = 0.

10 (SincefY(y) = fX,Y(x,y)dxandfX,Y 0,theright-handsideoftheab oveequationwillalsob e0atsuchy.) osethatXandYareindep endentand satis esE| (X,Y)|< .ThenE[ (X,Y)|X] =g(X)whereg(x) =E[ (x,Y)].Asinthepreviousexample,condition( i)issatis edbyDo ob-Dynkin,andcondition(ii)canb everi edbyletting and denotethedistributionsofXandY,resp ectively,andcomputing {X B}g(X)dP= Bg(x)d (x) = 1B(x)( (x,y)d (y))d (x)= 1B(x) (x,y)d( )(x,y) = 1B(X) (X,Y)dP= {X B} (X,Y) ertiesofordinaryexp ectationcarryovertoconditionalexp ectationastheyareultimatelyfactsab outintegrals:Prop (Linearity).E[aX+Y|G] =aE[X|G] +E[Y|G] ,andforanyA G A(aE[X|G] +E[Y|G])dP=a AE[X|G]dP+ AE[Y|G]dP=a AXdP+ AY dP= A(aX+Y)dP.