概率论与数理统计公式(Probabilitytheoryandmathematicalstatisticsformula)

1、概率论与数理统计公式（Probabilitytheoryandmathematicalstatisticsformula）Thefirstchapterstochasticeventsandtheirprobabilitypermutationsandcombinationsformulasareusedtopickoutthepossiblenumberofpermutationsofnindividualsfrommindividuals.Thenumberofpossiblecombinationsofnindividualsselectedfrommindividuals.(2)add

2、itionandmultiplicationprincipleadditionprinciple(twomethodscancompletethematter):m+nThetwomethodcanbeusedtocompleteacertainsubject.ThefirstmethodcanbecompletedbyMmethods,andthesecondmethodcanbecompletedbynmethods.Then,thismethodcanbecompletedbym+nmethods.Multiplicationprinciple(twostepscannotdothiss

3、eparately):mxnThefirststepcanbecompletedbyMmethods,andthesecondstepcanbecompletedbynmethods,andthiscanbeaccomplishedbym*nmethodsintwoways.(3)somecommonpermutationsarerepetitiveandnonrepetitive(ordered)Oppositeevents(atleastone)Orderproblem(4)randomtestandrandomeventsifatestisrepeatedinthesameconditi

4、ons,andeachtimethetestresultsmaybemorethanone,butbeforeatestisnottoassertthatitappearswhichresults,saidthestudywasarandomizedtrial.Thepossibleoutcomeoftheexperimentiscalledarandomevent.(5)basicevents,samplespacesandeventsinatest,regardlessofthenumberofevents,canalwaysfindsuchagroupofevents,ithasthef

5、ollowingproperties:Eachtrialmustoccurandonlyoneeventinthisgroupoccurs;Anyeventismadeupofsomeoftheeventsinthisgroup.Eacheventinsuchagroupofeventsiscalledabasicevent,whichisusedtorepresenttheevent.Thewholeofthebasiceventiscalledthesamplespaceofthetest.AneventisacollectionoflettersA,B,Careusuallyparts(

6、basicevents)used,.Representinginit.Capitalevents,theyaresubsets.Isitaninevitableevent,animpossibleone?.Theprobabilityofanimpossibleeventiszero,andtheeventwithzeroprobabilityisnotnecessarilyanimpossibleevent;similarly,theprobabilityoftheinevitableevent(omega)is1,andtheeventwithprobability1isnotnecess

7、arilyaninevitableevent.(6)therelationshipbetweeneventsandoperations:IfthecomponentoftheeventAisalsoapartoftheeventB,(Ahappens,theremustbeaneventB):Ifthereisasimultaneousevent,theeventAisequivalenttotheeventB,orAequalsB:A=B.ThereisatleastoneeventinAandB:AB,orA+B.AneventthatispartofAratherthanBiscalle

8、dthedifferencebetweenAandB,denotedasA-B,andcanalsobedenotedasA-AB,oritrepresentstheeventthatBdoesnothappenwhenAoccurs.AandBoccursimultaneously:A,B,orAB.AB=?,whichmeansthatAandBcannothappenatthesametime,calledeventAincompatiblewitheventBormutuallyexclusive.Basiceventsareincompatible.-Aiscalledtheinve

9、rseeventofeventA,ortheoppositeeventofA.ItrepresentsaneventthatdoesnotoccurinA.Mutualexclusionisnotnecessarilyopposite.Operations:Bindingrate:A(BC)=(AB)C,A(B,C)=(A,B),CThedistributionrate(AB),C=(A,C)a(B,C)(A,B)C=(AC),(BC)Therateofprobability:(7)theaxiomaticdefinitionofprobabilityissetasasamplespace.F

10、orevents,thereisarealnumberP(A)foreachevent,ifthefollowingthreeconditionsaresatisfied:10=P(A=1),2degreeP(omega)=13degreesfor22incompatibleevents,.YesItisoftencalledcountable(complete)additivity.P(A)iscalledtheprobabilityofevents.theclassicalprobabilitymodelis1degrees,2degree.Setanyevent,itismadeupof

11、,thereisP(A)=geometricprobabilityiftherandomtestresultsforinfiniteuncountableandeachresultsthepossibilityofuniform,andeverybasiceventinthesamplespacecanbeusedtodescribeaboundedregion,saidthetestforrandomgeometricprobability.Aforanyevent,.Lisgeometricmeasure(length,area,volume).(10)additiveformulaP(A

12、+B)=P(A)+P(B)-P(AB)WhenP(AB)=0,P(A+B)=P(A)+P(B)subtractionformulaP(A-B)=P(A)-P(AB)WhenBA,P(A-B)=P(A)-P(B)WhenA=,P()=1-P(B)(12)conditionalprobabilitydefinesAandBaretwoevents,andP(A)0iscalledtheconditionalprobabilityofeventBoccurringineventA.Conditionalprobabilityisakindofprobability,andallprobability

13、propertiesaresuitableforconditionalprobability.Forexample,P(omega/B)=1P(/A)=1-P(B/A)(13)multiplicationformulamultiplicationformula:Moregenerally,foreventA1,A2,.An,ifP(A1A2.An-1)0,butthereis(14)independence:theindependenceofthetwoeventsEventandsatisfactionarecalledevents,andtheyareindependentofeachot

14、her.Ifeventsaremutuallyindependent,andthenthereareIfeventsareindependentofeachother,theycanbeseparatedfromeachother.Inevitableeventsandimpossibleeventsareindependentofanyevent.Ismutuallyexclusivetoanyevent.TheindependenceofmultipleeventsLetABCbethreeevents,if22independentconditionsaresatisfied,P(AB)

15、=P(A)P(B);P(BC)=P(B)P(C);P(CA)=P(C)P(A)AndsatisfyP(ABC)=P(A)P(B)P(C)atthesametimeSoA,B,Careindependentofeachother.Similartonevents.allprobabilityformulasetseventsatisfaction1degrees22incompatibleeachother,2degree,Isthere.(16)Biasformulasetevent,.Andsatisfaction1degree,.22incompatible,0,1,2,.，degree,

16、beI=1,2,.N.ThisformulaistheBayesformula.(,.Aprioriprobability.(,.Itisusuallycalledposteriorprobability.TheBiasformulareflectsthecausalprobabilitylaw,andmadebyShuoyinfruitinference.(17)wehavedoneatestonBernoullishypothesisEachtrialhasonlytwopossibleoutcomesthatoccurordonotoccur;Thesecondarytestisrepe

17、ated,i.e.,theprobabilityofoccurrenceishomogeneousateachtime;Eachtrialisindependent,thatis,whethereachtrialoccursornotdoesnotaffecttheoccurrenceofothertrials.ThisexperimentiscalledtheBernoullimodel,ortheheavyBernoullitest.Theprobabilityofoccurrenceofeachtrialisexpressedastheprobabilityoftheoccurrence

18、ofthesecondintheheavyBernoullitest,，。ThesecondchapterrandomvariableanditsdistributiontheprobabilityofdiscreterandomvariableisXk(k=1,2),.Andtaketheprobabilityofeachvalue,thatis,theprobabilityoftheevent(X=Xk)isP(X=xk)=pk,k=1,2,.,Theupperboundistheprobabilitydistributionordistributionlawofdiscreterando

19、mvariables.Itissometimesgivenintheformofdistributedcolumns:.Obviously,thedistributionlawshouldmeetthefollowingconditions:(1),(2).(2)thedistributiondensityofcontinuousrandomvariablesisthedistributionfunctionofrandomvariables.Ifthereisanonnegativefunction,thereisanarbitraryrealnumber,Itiscalledcontinu

20、ousrandomvariable.Probabilitydensityfunctionordensityfunction,referredtoasprobabilitydensity.Thedensityfunctionhasthefollowing4properties:degree.degree.therelationshipbetweendiscreteandcontinuousrandomvariablesThefunctionoftheintegralelementinthetheoryofcontinuousrandomvariableissimilartothatintheth

21、eoryofdiscreterandomvariables.thedistributionfunctionisarandomvariable,anditisanarbitraryrealnumberThedistributionfunctionofX,arandomvariable,isessentiallyacumulativefunction.YoucangettheprobabilitythatXfallsintotherange.Thedistributionfunctionrepresentstheprobabilityoftherandomvariablefallingintoth

22、einterval(-,x).Thedistributionfunctionhasthefollowingproperties:1degree;2degreesaremonotonedecreasingfunctions;degree,;degrees,thatis,rightcontinuous;degree.Fordiscreterandomvariables,;Forcontinuousrandomvariables,.(5)eightdistribution,0-1distributionP(X=1)=p,P(X=0)=qIntheNouritest,thetwodistributio

23、nistheprobabilityofeventoccurrence.Thenumberofeventsisarandomvariable,andifitis,itmaybevaluedas.Amongthem,Itiscalledthetwodistributionofrandomvariablesobeyingtheparameter.Rememberas.Atthattime,thisis(0-1)distribution,so(0-1)thedistributionisaspecialcaseofthetwodistribution.Thedistributionlawofrandom

24、variablesisgivenbyPoissondistribution，ThePoissondistribution,whichiscalledtherandomparameter,isdenotedasorP().ThePoissondistributionisthelimitdistributionofthetwoterms(np=,N,P).HypergeometricdistributionThehypergeometricdistributionoftherandomvariableXfollowstheparametern,N,M,denotedbyH(n,N,M).Thege

25、ometricdistribution,whereinP=0,q=1-p.ThegeometricdistributionoftherandomvariableXobeyingtheparameterpisdenotedasG(P).Thevalueoftherandomvariableisonlya,b,andthedensityfunctionisconstantonaandbOther，Therandomvariableisuniformlydistributedonaandb,andisdenotedasXU(a,B).ThedistributionfunctionisWhena=x1

26、x2=B,Xfallsintherangeof()inprobability.exponentialdistributionAmongthem,theexponentialdistributionoftherandomvariableXobeystheparameter.ThedistributionfunctionofXisRemembertheintegralformula:Thedensityfunctionofrandomvariablesisnormaldistribution，Wheretheconstantiscalledtherandomvariable,thenormaldi

27、stributionorGauss(Gauss)distributionisassumedastheparameter.Ithasthefollowingproperties:Thefigureof1degreesisaboutsymmetry;At2degrees,themaximumvaluewasthen;If,thenthedistributionfunctionis.Thenormaldistributionofparametersandtimeiscalledthenormalnormaldistribution,andthedensityfunctioniscalledasthe

28、normaldistribution，Thedistributionfunctionis.Itisanonintegralfunction,anditsfunctionvaluehasbeencompiledasatableforreference.(-x)=1-(x)and(0)=.If,then.(6)quantiletable:;Upperquartiletable:.(7)thediscretedistributionofthefunctiondistributionisknownasthedistributionoftheknowndistribution,Thedistributi

29、oncolumns(notequal)areasfollows:,Ifthereissomeequality,theprobabilityoftheadditionofthecorrespondingsumshouldbeconsidered.ByusingcontinuousprobabilitydensityfXX(x)towritethedistributionfunctionFY(y)Y=P(g(X)=y,fY)iscalculatedusingthederivationformulaofvariableupperlimitintegral(Y).Thethirdchaptertwod

30、imensionalrandomvariableanditsdistribution(1)ifthejointdistributionisdiscrete,ifallpossiblevaluesoftwo-dimensionalrandomvectors(X,Y)areatmostcountableorderedpairs(x,y),thentheyarecalleddiscreterandomvariables.Let=(X,Y)haveallpossiblevalues,andtheprobabilityofevent=isPIJ,calledThedistributionlawof=(X

31、,Y)isalsocalledthejointdistributionlawofXandY.Thejointdistributionissometimesrepresentedbythefollowingprobabilitydistributiontable:YXY1y2.Yj.X1P11p12.P1j.X2p21p22.P2j.Xipi1.Here,PIJhasthefollowingtwoproperties:PIJ=0(I,j=1,2,.);continuousfortwo-dimensionalrandomvector,ifthereisanonnegativefunction,so

32、thatanyoneofitsadjacentedgesareparalleltotheaxisoftherectangularregionD,thatis,D=(X,Y)|axb,cyx1,F(X2,y)=F(x1,y);wheny2y1,F(x,Y2)=F(x,Y1);F(x,y)isrightcontinuousforXandy,respectivelyfor.therelationshipbetweendiscreteandcontinuoustype(5)theedgedistributionofthediscreteXistheedgedistribution;Themargina

33、ldistributionofYis.TheedgedistributiondensityofcontinuousXisThemarginaldistributiondensityofYis0undertheconditionofknownX=xi,theconditionaldistributionofYvalueisdiscreteUndertheconditionofknownY=yj,theconditionaldistributionofXvalueisUndertheconditionofcontinuousY=y,theconditionaldistributiondensity

34、ofXis0;UndertheconditionofknownX=x,theconditionaldistributiondensityofYis0independentgeneraltypeF(X,Y)=FX(x)FY(y)DiscretetypeZeroindependenceContinuousf(x,y)=fX(x)fY(y)Directjudgment,sufficientandnecessarycondition:SeparablevariablesTheintervalofpositiveprobabilitydensityisrectangleTwodimensionalnor

35、maldistributionAfunctionof=0randomvariablesifX1,X2,.Xm,Xm+1,.Xnisindependentofeachother,handGarecontinuousfunctions:H(X1,X2),.Xm)andG(Xm+1),.Xn)independentofeachother.Specialcase:ifXandYareindependent,thenH(X)andG(Y)areindependent.Forexample,ifXandYareindependent,then3X+1and5Y-2areindependent.(8)the

36、distributiondensityfunctionofrandomvector(X,Y)istwodimensionaluniformdistributionWhereSDistheareaoftheregionD,itiscalled(X,Y)obeystheuniformdistributiononD,andisdenotedas(X,Y)U(D).Forexample,Figure3.1,Figure3.2andfigure3.3.YOneD1O1xFigure3.1YOneO2xFigure3.2YDCOaBxFigure3.3thedistributionfunctionofth

37、erandomvector(X,Y)istwodimensionalnormaldistributionAmongthem,5parametersarecalled(X,Y)andobeytwodimensionalnormaldistribution,Itisdenotedas(X,Y)NFromthecalculationformulaofedgedensity,itcanbededucedthatthetwonormaldistributionofthetwodimensionalnormaldistributionisstillnormaldistribution,Thatis,XNH

38、owever,ifXN(X,Y)isnotnecessarilyatwo-dimensionalnormaldistribution).(10)functiondistributionZ=X+Yiscalculatedaccordingtodefinition:Forthecontinuoustype,fZ(z)=Twoindependentnormaldistributionsandstillnormaldistribution().Thelinearcombinationofnindependentnormaldistributionstillobeysthenormaldistribut

39、ion.Z=max,min(X1,X2),.Xn)iftheyareindependentofeachother,theirdistributionfunctionsareZ=max,min(X1,X2),.ThedistributionfunctionofXnis:distributionIfnrandomvariablesareindependentofeachotherandobeythestandardnormaldistribution,thesumofsquaresofthemcanbeprovedThedistributiondensityis0Wecalltherandomva

40、riableWtoobeythedistributionofdegreeoffreedomn,denotedasWTheso-calleddegreeoffreedomreferstothenumberofindependentnormalrandomvariables,whichisanimportantparameterinthedistributionofrandomvariables.Thedistributionsatisfiesadditivity:SetbeThetdistributionisX,andYistwoindependentrandomvariablesFunctio

41、ncanbeprovedTheprobabilitydensityis0WecalltherandomvariableTtoobeythetdistributionofdegreeoffreedomn,denotedasTt(n).Fdistributionisset,andXandYareindependent,andtheprobabilitydensityfunctioncanbeprovedtobeWecalltherandomvariableFobeytheFdistributionwiththefirstdegreeoffreedomasN1andtheseconddegreeof

42、freedomasN2,whichisdenotedasFf(N1,N2)Thefourthchapteristhenumericalcharacteristicsofrandomvariablesdiscretecontinuoustypeofdigitalcharacteristicsofone-dimensionalrandomvariablesExpectExpectationisthemeanvalue,andXisadiscreterandomvariable,whosedistributionlawisP()=PK,k=1,2,.N,(absoluteconvergence)Xi

43、sacontinuousrandomvariablewhoseprobabilitydensityisf(x),(absoluteconvergenceisrequired)TheexpectedY=g(X)offunctionY=g(X)varianceD(X)=EX-E(X)2,standarddeviation,Forthepositiveintegerk,themathematicalexpectationoftheKpoweroftherandomvariableXisthekorderoriginmomentofX,whichisdenotedasVKVk=E(Xk=k=1,2),

44、.Forthepositiveintegerk,themathematicalexpectationoftheKpoweroftherandomvariableXandE(X)differenceisthecentralmomentofkorderofX=,k=1,2,.Forthepositiveintegerk,themathematicalexpectationoftheKpoweroftherandomvariableXisthekorderoriginmomentofX,whichisdenotedasVKV(Xk)=k=EK=1,2,.Forthepositiveintegerk,

45、themathematicalexpectationoftheKpoweroftherandomvariableXandE(X)differenceisthecentralmomentofkorderofX=K=1,2,.TheChebyshevinequalityhasarandomvariableXwithmathematicalexpectationE(X)=a,varianceD(X)=sigma2,thenforanypositivenumberepsilon,therearethefollowingChebyshevinequalitiesTheChebyshevinequalit

46、ygivestheprobabilityinthecaseofthedistributionofunknownXItisofgreatsignificanceintheory.thepropertiesofexpectation(1)E(C)=CE(CX)=CE(X)E(X+Y)=E(X)+E(Y),E(XY)=E(X)E(Y),sufficientcondition:XandYareindependent;Necessaryandsufficientconditions:XandYareuncorrelated.thenatureofvariance(1)D(C)=0;E(C)=CD(aX)

47、=a2D(X);E(aX)=aE(X)D(aX+b)=a2D(X);E(aX+b)=aE(X)+bD(X)=E(X2)-E2(X)D(X+Y)=D(X)+D(Y),sufficientcondition:XandYareindependent;Necessaryandsufficientconditions:XandYareuncorrelated.D(X+Y)=D(X)+D(Y)+2E(X-E(X)(Y-E(Y)isunconditionallyestablished.AndE(X+Y)=E(X)+E(Y)isunconditionallyestablished.expectationand

48、varianceexpectationvarianceofcommondistribution0-1distributionPtwoitemdistributionNPPoissondistributionGeometricdistributionHypergeometricdistributionuniformdistributionexponentialdistributionNormaldistributionN2nDigitalcharacteristicexpectationoftdistribution0(n2)(5)twodimensionalrandomvariablesExp

49、ectationoffunction=varianceCovarianceforrandomvariablesXandY,calledtheirtwoordermixedcentralmomentsforXandYcovarianceorcorrelationmoments,thatis,thatisCorrespondingtothesign,thevarianceD(X)andD(Y)ofXandYcanalsobedenotedasand.ThecorrelationcoefficientsforrandomvariablesXandY,ifD(X)0,D(Y)0,arecalledAs

50、thecorrelationcoefficientbetweenXandY,denotedby(sometimesabbreviatedas).|islessthan1,when|=1,XandYcompletely:CompletecorrelationAtthetime,XwasnotassociatedwithY.Thefollowingfivepropositionsareequivalent:1;CoV(X,Y)=0;E(XY)=E(X)E(Y);D(X+Y)=D(X)+D(Y);D(X-Y)=D(X)+D(Y)Themixedmomentsofcovariancematrixfor

51、randomvariablesXandY,ifexist,arecalledXandYmixedordermomentsofk+lorder,denotedask+lmixedcentralmoment:thenatureofcovariance(I)cov(X,Y)=cov(Y,X);(II)cov(aX,bY)=abcov(X,Y);(III)cov(X1+X2,Y)=cov(X1,Y)+cov(X2,Y);(IV)cov(X,Y)=E(XY)-E(X)E(Y)independentanduncorrelated(I),iftherandomvariableXandYareindepend

52、entofeachother,then,conversely,nottrue.(II)if(X,Y)N(),ThenecessaryandsufficientconditionfortheindependenceofXandYisthatXandYareuncorrelated.TheFifthLawoflargenumbersandthecentrallimittheorem(1)thelawoflargenumbersChebyshevslawoflargenumberswithrandomvariablesX1,X2,.Theyareindependentofeachotherandha

53、vefinitevariance,andareboundedbythesameconstantC:D(Xi)C(i=1,2,.Foranypositivenumberepsilon,thereisapositivenumberSpecialcase:ifX1,X2,.WiththesamemathematicalexpectationE(XI)=muon,thentheupperformbecomes.BernoullislawoflargenumbersisthenumberofeventsAoccurringinthenindependenttest,andPistheprobabilit

54、ythattheeventAoccursineachtest,andforanypositivenumberepsilon,thereisBernoullislawoflargenumbersshowsthatwhenthenumberoftestnislarge,theprobabilityofoccurrenceofeventAismorelikelytobedistinguishedThisdescribesthestabilityofthefrequencyinstrictmathematicalform.ThelawoflargenumbersforsymplecticsetX1,X

55、2,.Xn,.Itisasequenceofindependentandidenticallydistributedrandomvariables,andE(Xn)=(2)centrallimittheoremLeviLindbergtheoremlettherandomvariableX1,X2,.Theyareindependentofeachother,obeythesamedistribution,havethesamemathematicalexpectationandvariance:then,randomvariableandtheThedistributionfunctionF

56、n(x)hasanarbitraryrealnumberxThistheoremisalsocalledthecentrallimittheoremofindependenceandidenticallydistribution.LaplassedeMoivretheoremletrandomvariableswithparametersN,P(0p1)ofthetwodistribution,thenforanyrealnumberx,a(3)ifthetwotheoremsarepositive,then.Thelimitdistributionofhypergeometricdistri

57、butionistwotermdistribution.ifthePoissontheoremispositive,thenAmongthem,k=0,1,2,.N,.ThelimitdistributionofthetwodistributionisPoissondistribution.Thesixthchapter:sampleandsamplingdistributionthebasicconceptofmathematicalstatisticsinthemathematicalstatistics,oftentheobjectofa(ormore)indicatorsofthewh

58、oleknownastheoverall(ormaternal).Wealwaysregardthepopulationasarandomvariable(orarandomvector)withdistribution.Eachunitintheindividualpopulationiscalledasample(oranindividual).Inthesample,wecallsomesamplesextractedfromthepopulationassamples.Thenumberofsamplescontainedinthesampleiscalledthesamplesize

59、,whichisgenerallyexpressedinn.Ingeneral,thesamplesareregardedasnindependentvariableswiththesamedistributionasthepopulation.Suchsamplesarecalledsimplerandomsamples.Whenreferringtotheresultsextractedatanytime,theyrepresentnrandomvariables(samples);afteraspecificdecimation,theyrepresentnspecificvalues(

60、samplevalues).Wecallitthedualityofthesample.Thesamplefunctionandstatisticsaresetasasampleofthepopulation()Forthesamplefunction,whichisacontinuousfunction.Ifanyunknownparameterisnotincluded,itiscalled(a)statistic.Commonstatisticsandtheirproperties,samplemeanSamplevarianceSamplestandarddeviationSample