概率论与数理统计英文总结材料

概率论与数理统计英文版总结资料概率论与数理统计英文版总结资料概率论与数理统计英文版总结资料合用标准SampleSpace样本空间Thesetofallpossibleoutcomesofastatisticalexperimentiscalledthesamplespace.Event事件Aneventisasubsetofasamplespace.certainevent（必然事件）：ThesamplespaceSitself,iscertainlyanevent,whichiscalledacertainevent,meansthatitalwaysoccursintheexperiment.impossibleevent（不可以能事件）：Theemptyset,denotedby,isalsoanevent,calledanimpossibleevent,meansthatitneveroccursintheexperiment.Probabilityofevents（概率）Ifthenumberofsuccessesinntrailsisdenotedbys,andifthesequenceofrelativefrequenciess/nobtainedforlargerandlargervalueofnapproachesalimit,thenthislimitisdefinedastheprobabilityofsuccessinasingletrial.“equallylikelytooccur”------probability（古典概率）IfasamplespaceSconsistsofNsamplepoints,eachisequallylikelytooccur.AssumethattheeventAconsistsofnsamplepoints,thentheprobabilitypthatAoccursispP(A)

nNMutuallyexclusive(互斥事件)文档合用标准Definition2.4.1EventsA1,A2,L,Anarecalledmutuallyexclusive,ifAiIAj,ij.IfAandBaremutuallyexclusive,thenP(AUB)P(A)P(B)(2.4.1)Mutuallyindependent事件的独立性TwoeventsAandBaresaidtobeindependentifP(AIB)P(A)P(B)OrTwoeventsAandBareindependentifandonlyifP(B|A)P(B).ConditionalProbability条件概率Theprobabilityofaneventisfrequentlyinfluencedbyotherevents.DefinitionTheconditionalprobabilityofB,givenA,denotedbyP(B|A),isdefinedbyP(AIB)P(B|A)P(A)

ifP(A)0.(2.5.1)Themultiplicationtheorem乘法定理IfA1,A2,L,Akareevents,thenP(A1IA2ILAk)P(A1)P(A2|A1)P(A3|A1IA2)LP(Ak|A1IA2ILIAk1)IftheeventsA1,A2,L,Akareindependent,thenforanysubset{i1,i2,L,im}{1,2,L,k},P(Ai1IAi2ILAim)P(Ai1)P(Ai2)LP(Aim)文档合用标准(全概率公式totalprobability)Theorem2.6.1.IftheeventsB1,B2,L,BkconstituteapartitionofthesamplespaceSsuchthatP(Bj)0forj1,2,L,k,thanforanyeventAofS,kkP(A)P(AIBj)j1

P(Bj)P(AIBj)(2.6.2)j1（贝叶斯公式Bayes’formula.）Theorem2.6.2IftheeventsB1,B2,L,BkconstituteapartitionofthesamplespaceSsuchthatP(Bj)0forj1,2,L,k,thanforanyeventAofS,P(A)0,P(Bi|A)P(Bi)P(A|Bi).fori1,2,L,kkP(Bj)P(A|Bj)j1(2.6.2)ProofBythedefinitionofconditionalprobability,P(BiIA)P(Bi|A)P(A)Usingthetheoremoftotalprobability,wehaveP(Bi|A)kP(Bi)P(A|Bi)i1,2,L,kP(Bj)P(A|Bj)1randomvariabledefinition文档合用标准Arandomvariableisarealvaluedfunctiondefinedonasamplespace;i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.2.DistributionfunctionLetXbearandomvariableonthesamplespaceS.ThenthefunctionF(X)P(Xx).xRiscalledthedistributionfunctionofXNoteThedistributionfunctionF(X)isdefinedonrealnumbers,notonsamplespace.PropertiesThedistributionfunctionF(x)ofarandomvariableXhasthefollowingproperties:F(x)isnon-decreasing.Infact,ifx1x2,thentheevent{Xx1}isasubsetoftheevent{Xx2},thusF(x1)P(Xx1)P(Xx2)F(x2)(2)F( )limF(x)0,xF( )limF(x)1.x(3)Foranyx0R,limF(x)F(x00)F(x0).Thisistosay,thexx00distributionfunctionF(x)ofarandomvariableXisrightcontinuous.文档合用标准3.2DiscreteRandomVariables失散型随机量DefinitionArandomvariableXiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequence{a1,a2,Lan,L}geometricdistribution（几何分布）X1234⋯k⋯Ppq1q2q3qk－ppp1p⋯Binomialdistribution（二项分布）DefinitionThenumberXofsuccessesinnBernoullitrialsiscalledabinomialrandomvariable.Theprobabilitydistributionofthisdiscreterandomvariableiscalledthebinomialdistributionwithparametersnandp,denotedbyB(n,p).poissondistribution（泊松分布）DefinitionAdiscreterandomvariableXiscalledaPoissonrandomvariable,ifittakesvaluesfromtheset{0,1,2,L},andifk,P(Xk)p(k;)e0k0,1,2,Lk!(3.5.1)Distribution(3.5.1)iscalledthePoissondistributionwith文档合用标准parameter,denotedbyP().Expectation(mean)数学希望DefinitionLetXbeadiscreterandomvariable.TheexpectationormeanofXisdefinedasE(X)xP(Xx)(3.3.1)x2．Variance方差standarddeviation（标准差）LetXbeadiscreterandomvariable,havingexpectationE(X).ThenthevarianceofX,denotebyD(X)isdefinedastheexpectationoftherandomvariable(X)2D(X)E(X)2(3.3.6)ThesquarerootofthevarianceD(X),denotebyD(X),21iscalledthestandarddeviationofX:D(X)EX2(3.3.7)probabilitydensityfunction概率密度函数DefinitionAfunctionf(x)definedon(,)iscalledaprobabilitydensityfunction（概率密度函数）if:f(x)0foranyxR;(ii)f(x)isintergrable(可积的)on(,)andf(x)dx1.文档合用标准Letf(x)beaprobabilitydensityfunction.IfXisarandomvariablehavingdistributionfunctionxF(x)P(Xx)f(t)dt,(4.1.1)thenXiscalledacontinuousrandomvariablehavingdensityfunctionf(x).Inthiscase,x2P(x1Xx2)f(t)dt.(4.1.2)x15.Mean（均值）Definition4.1.2LetXbeacontinuousrandomvariablehavingprobabilitydensityfunctionf(x).Thenthemean(orexpectation)ofXisdefinedbyE(X)xf(x)dx,(4.1.3)6.variance方差Similarly,thevarianceandstandarddeviationofacontinuousrandomvariableXisdefinedbyD(X)E((X)2),(4.1.4)WhereE(X)isthemeanofX,isreferredtoasthestandarddeviation.文档合用标准Weeasilyget2D(X)x2f(x)dx2.(4.1.5).4.2UniformDistribution均匀分布Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunction1foraxb,f(x)ba0elsewhere,4.5ExponentialDistribution指数分布Definition4.5.1AcontinuousvariableXhasanexponentialdistributionwithparameter(0),ifitsdensityfunctionisgivenby1exforx0f(x)(4.5.1)0forx0ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenbyE(X),D(X)2.文档合用标准4.3NormalDistribution正态分布1.DefinitionTheequationofthenormalprobabilitydensity,whosegraphisshowninFigure4.3.1,isf(x)1e(x)2/22x24.4NormalApproximationtotheBinomialDistribution（二项分布）X~B(n,p),nislarge(n>30),piscloseto0.50,X~B(n,p)N(np,npq)4.7Chebyshev’sTheorem（切比雪夫定理）Ifaprobabilitydistributionhasmeanμandstandarddeviationσ,theprobabilityofgettingavaluewhichdeviatesfromμbyatleastkσisatmost1k2.Symbolically,P(|X|k)1k2.Jointprobabilitydistribution（联合分布）Inthestudyofprobability,givenatleasttworandomvariablesX,Y,...,thataredefinedonaprobabilityspace,thejointprobabilitydistributionforX,Y,...isaprobability文档合用标准distributionthatgivestheprobabilitythateachofX,Y,...fallsinanyparticularrangeordiscretesetofvaluesspecifiedforthatvariable.Conditionaldistribution条件分布ConsistentwiththedefinitionofconditionalprobabilityofeventswhenAistheeventX=xandBistheeventY=y,theconditionalprobabilitydistributionofXgivenY=yisdefinedaspX(x|y)p(x,y)forallxprovidedpY(y)

pY(y)0.Statisticalindependent随机变量的独立性Definition5.3.1Supposethepair{X,Y}ofrealrandomvariableshasjointdistributionfunctionF(x,y).IftheF(x,y)obeytheproductruleF(x,y)FX(x)FY(y)forallx,y.thetworandomvariablesXandYareindependent,orthepair{X,Y}isindependent.5.4CovarianceandCorrelation协方差和相关系数WenowdefinetworelatedquantitieswhoseroleincharacterizingtheinterdependenceofXandYwewanttoexamine.Definition5.4.1SupposeXandYarerandomvariables.Thecovarianceofthepair{X,Y}isCov(X,Y)E[(XX)(YY)].Thecorrelationcoefficientofthepair{X,Y}isCov(X,Y)(X,Y).XYWhereXE(X),YE(Y),XD(X),YD(Y).文档合用标准Definition5.4.2TherandomvariablesXandYaresaidtobeuncorrelatediffCov(X,Y)0.中心5.5LawofLargeNumbersandCentralLimitTheorem极限制理Wecanfindthesteadilyofthefrequencyoftheeventsinlargenumberofrandomphenomenon.Andtheaverageoflargenumberofrandomvariablesarealsosteadiness.Theseresultsarethelawoflargenumbers.Ifasequenceindependent,with

{Xn:n1}ofrandomvariablesisE(Xn),D(Xn)2,thenlimP(|1nXk|)1,forany0.(5.5.1)nnk1LetnAequalsthenumberoftheeventAinnBernoullitrials,andpistheprobabilityoftheeventAonanyoneBernoullitrial,thenlimP(|nA|)1forany0.(5.5.2)nn(频率拥有牢固性)IfXn(n1)isindependent,withE(Xn),D(Xn)2,andSn*SnnnthenlimFn(x)(x)forallx.x文档合用标准population(整体).Definition6.2.1Apopulationisthesetofdataormeasurementsconsistsofallconceivablypossibleobservationsfromallobjectsinagivenphenomenon.Apopulationmayconsistoffinitelyorinfinitelymanyvarieties.sample（样本、子样）Definition6.2.2Asampleisasubsetofthepopulationfromwhichsampling(抽样)peoplecandrawconclusionsaboutthewhole.takingasample:Theprocessofperforminganexperimenttoobtainasamplefromthepopulationiscalledsampling.中位数DefinitionIfarandomsamplehastheorderstatisticsX(1),X(2),,X(n),then(i)TheSampleMedianisXn1)ifnisodd(M021XXnifnisevenn(1))((ii)TheSampleRangeisRX(n)X(1).SampleDistributions抽样分布文档合用标准1．samplingdistributionofthemean均值的抽样分布Theorem6.3.1IfXismeanoftherandomsampleX1,X2,,XnofsizenfromarandomvariableXwhichhasmeanandthevariance2,then2E(X)andD(X).nItiscustomarytowriteE(X)asXandD(X)asHere,E(X)iscalledtheexpectationofthemean望iscalledthestandarderrorofthemean.Xn

2X..均值的期均值的标准差PointEstimate点估计DefinitionSupposeisaparameterofapopulation,X1,,Xnisarandomsamplefromthispopulation,andT(X1,,Xn)isastatisticthatisafunctionofX1,Xn.Now,totheobservedvaluex1,,xn,ifweuseT(x1,,xn)asanestimatedvalueof,thenT(X1,,Xn)iscalledapointestimatorofandT(x1,,xn)isreferredasapointestimateof.Thepointestimatorisalsooftenwrittenas?.Unbiasedestimator(无偏估计量)Definition7.1.2.Suppose?isanestimatorofaparameter.Then?isunbiasedifandonlyif文档合用标准E(?).minimumvarianceunbiasedestimator（最小方差无偏估计量）Let?beanunbiasedestimatorof.Ifforany?'whichisalsoanunbiasedestimatorof,wehaveD(?)D(?'),then?iscalledtheminimumvarianceunbiasedestimatorof.Sometimesitisalsocalledbestunbiasedestimator.3.MethodofMoments矩估计的方法SupposeX1,X2,,XnconstitutearandomsamplefromthepopulationXthathaskunknownparameters1,2,,k.Also,thepopulationhasfirskfinitemomentsE(X),E(X2),,E(Xk)thatdependsontheunknownparameters.E(X)E(X2)E(Xk)

1nXini11nXi2ni1,(7.1.4)1nkniXi1togetunknownparametersexpressedbytheobservationsvalues,i.e.j?j(X1,X2,,Xk)forj1,2,,k.Then?jisanestimatorofjbymethodofmoments.文档合用标准Supposethatisaparameterofapopulation,X1,,Xnisarandomsampleoffromthispopulation,and?T1(X1,,Xn)and?T2(X1,,Xn)aretwo12statisticssuchthat??.Ifforagivenwith0,we1haveP(?1?2)1.Thenwereferto[?1,?2]asa(1)100%confidenceintervalfor.Moreover,1iscalledthedegreeofconfidence.?1and?2arecalledlowerandupperconfidencelimits.Theestimationusingconfidenceintervaliscalledintervalestimation.confidenceinterval-----置信区间lowerconfidencelimits-----置信下限upperconfidencelimits-----置信上限deg