概率论与数理统计英文版总结.docx

SampleSpace样本空间Thesetofallpossibleoutcomesofastatisticalexperimentiscalledthesamplespace.Event事件Aneventisasubsetofasamplespace.certainevent(必然事件)：ThesamplespaceSitself,iscertainlyanevent,whichiscalledacertainevent,meansthatitalwaysoccursintheexperiment.impossibleevent(不可能事件)：Theemptyset,denotedby0,isalsoanevent,calledanimpossibleevent,meansthatitneveroccursintheexperiment.Probabilityofevents(概率)Ifthenumberofsuccessesinntrailsisdenotedbys,andifthesequenceofrelativefrequenciess/nobtainedforlargerandlargervalueofnapproachesalimit,thenthislimitisdefinedastheprobabilityofsuccessinasingletrial.equallylikelytooccur”probability(古典概率)IfasamplespaceSconsistsofNsamplepoints,eachisequallylikelytooccur.AssumethattheeventAconsistsofnsamplepoints,thentheprobabilityPthatAoccursisMutuallyexclusive(互斥事件)Definition2.4.1EventsA,A2,Aarecalledmutuallyexclusive,ifAiAj=0,Xfij.Theorem2.4.1IfAandBaremutuallyexclusive,thenP(AB)=P（八）+P(B)(2.4.1)Mutuallyindependent事件的独立性TwoeventsAandBaresaidtobeindependentifOrTwoeventsAandBareindependentifandonlyifP(BA)=P(B).ConditionalProbability条件概率Theprobabilityofaneventisfrequentlyinfluencedbyotherevents.DefinitionTheconditionalprobabilityofB,givenA,denotedbyP(3A),isdefinedbyP(BlA)=P(；(AFifP（八）>0.(2.5.1)Themultiplicationtheorem乘法定理If142,Aareevents,thenIftheeventsl42,4kareindependent,thenforanysubsetzl2,1,2,(全概率公式totalprobability)Theorem2.6.1.IftheeventsB1,B2,纥constituteapartitionofthesamplespaceSsuchthatP(Bj)0forj=1,2,k,thanforanyeventAOfS,kkP（八）=2P(ABj)=£P(Bj)P(ABj)(2.6.2)j=lj=(贝叶斯公式Bayes,formula.)IftheeventsB1,B2,BkconstituteapartitionofthesamplespaceSsuchthatP(Bj)Ofor)=1,2,k,thanforanyeventAofS,P（八）O,P(IlA)=/(耳)P(AIBLfOrf=1,2,yk(2.6.2)£P(Bj)P(AIJ)j=ProofBythedefinitionofconditionalprobability,Usingthetheoremoftotalprobability,Wehave1. randomvariabledefinitionArandomvariableisarealvaluedfunctiondefinedonasamplespace;1 .e.itassignsarealnumbertoeachsamplepointinthesamplespace.Distributionfunction1.etXbearandomvariableonthesamplespaceS.ThenthefunctionF(X)=P(Xx).xeRiscalledthedistributionfunctionofXNoteThedistributionfunctionF(X)isdefinedonrealnumbers,notonsamplespace.2. PropertiesThedistributionfunctionF(x)ofarandomvariableXhasthefollowingproperties:(l)F(x)isnon-decreasing.Infact,ifl2,thentheeventX<1isasubsetoftheeventX<x2,thus(2)F()=IimF(x)=0,x-oo产(+oo)=IimF(x)=1.(3)ForanyXqWR,IimF(x)=F+0)=F(x0).Thisistosay,thedistribution.t.¾+0uufunctionF(x)ofarandomvariableXisrightcontinuous.3.2DiscreteRandomVariables离散型随机变量Definition3.2.1ArandomvariableXiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequenceal9a2,an,geometricdistribution(几何分布)X1234kPPq1pq2pq3pk1qpBinomialdistribution(二项分布)ThenumberXofsuccessesinnBernoullitrialsiscalledabinomialrandomvariable.TheprobabilitydistributionofthisdiscreterandomvariableiscalledthebinomialdistributionwithparametersnandP,denotedbyB(,p).poissondistribution(泊松分布)Definition3.5.1AdiscreterandomvariableXiscalledaPoissonrandomvariable,ifittakesvaluesfromtheset0,l,2,andifP(X=k)=p(K)=一,>OZ=O,1,2,k1) .5.1)Distribution(3.5.1)iscalledthePoissondistributionwithparameter,denotedbyP().Expectation(mean)数学期望Definition3.3.1LetXbeadiscreterandomvariable.TheexpectationormeanofXisdefinedas=E(X)=ZxP(X=JV)(3.3.1)X2) Variancestandarddeviation(标准差)Definition3.3.2LetXbeadiscreterandomvariable,havingexpectationE(X)=.ThenthevarianceofX,denotebyD(X)isdefinedastheexpectationoftherandomvariable(X-/)2O(X)=E(X-M2)(3.3.6)ThesquarerootofthevarianceD(X),denotebyJo(X),is£calledthestandarddeviationofX:yD(X)=(eX-)2V(3.3.7)probabilitydensityfunction概率密度函数Afunctiondefinedon(-oo,)iscalledaprobabilitydensityfunctionIS率密度函数)if:(i)f(x)OforanyxeR；OO(h)fi,x)isintergrable(可积的)on(-,)andJf(x)dx=1.-CODefinition4.1.21.et(x)beaprobabilitydensityfunction.IfXisarandomvariablehavingdistribulionfunctionF(x)=P(Xx)=ftdt,(4.1.1)-00thenXiscalledacontinuousrandomvariablehavingdensityfunctionX).Inthiscase,P(x1<X<x2)=ft)dt.(4.1.2)x5.Mean(均值)LetXbeacontinuousrandomvariablehavingprobabilitydensityfunctionf(x).Thenthemean(orexpectation)ofXisdefinedby=E(X)=Jxf(x)dx,(4.1.3)variance方差-OOSimilarly,thevarianceandstandarddeviationofacontinuousrandomvariableXisdefinedby2=D(X)=E(X-)2)i(4.1.4)Where=E(X)isthemeanofX,isreferredtoasthestandarddeviation.Weeasilyget002=D(X)=x1fxdx-1.(4.1.5)4.2UniformDistribution均匀分布Theuniformdistribution,with(heparametersCmdb,hasprobabilitydensityfunction4.5ExponentialDistribution指数分布AcontinuousvariableXhasanexponentialdistributionwithparameter(>0),ifitsdensityfunctionisgivenby1W、eforx>0fM=(4.5.1)0forx0ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenbyE(X)=九D(X)=A2.4.3 NormalDistribution正态分布1. DefinitionTheequationofthenormalprobabilitydensity,whosegraphisshowninFigure4.3.1,is4.4 NormalApproximationtotheBinomialDistribution(二项分布)XB(n,p),nislarge(n>30),piscloseto0.50,4.7Chebyshev,sTheorem(切比雪夫定理)Ifaprobabilitydistributionhasmeanandstandarddeviation,theprobabilityofgettingavaluewhichdeviatesfrombyatleastkisatmost-.Symbolically,P(X-k)<.ICJointprobabilitydistribution(联合分布)Inthestudyofprobability,givenatleasttworandomvariablesX,Y,.,thataredefinedonaprobabilityspace,thejointprobabilitydistributionforX,Y,.isaprobabilitydistributionthatgivestheprobabilitythateachofX,Y,.fallsinanyparticularrangeordiscretesetofvaluesspecifiedforthatvariable.5.2Conditionaldistribution条件分布ConsistentwiththedefinitionofconditionalprobabilityofeventswhenAistheeventX=xandBistheeventY=y,theconditionalprobabilitydistributionofXgivenY=yisdefinedasPx(xy)="Cforallxprovidedp(y)0.p(y)5.3Statisticalindependent随机变量的独立性SupposethepairX,Yofrealrandomvariableshasjointdistributionfunctiony).IftheF(x,y)obeytheproductrule尸(x,y)=F(x)Fy(y)forallx,y.thetworandomvariablesXandYareindependent,orthepair(X,Yisindependent.theinterdependenceofXandYwewanttoexamine.SupposeXandYarerandomvariables.ThecovarianceofthepairX,YisTherandomvariablesXandYaresaidtobeUncorrelatcdiffCov(X,7)=0.ICc/YV、COV(X,Y)Ip=p(x,y)=-mberofWhereRX=E(X),=E(Y)ycx=JaX),o=Jay).blesarealsosteadiness.Theseresultsarethelawoflargenumbers.IfasequenceXnrnlofrandomvariablesisindependent,withthen1“IimP(VX-z<)=1,forany>O.(5.5.1)nA=1Theorem5.5.2Let11equalsthenumberoftheeventAinnBernoullitrials,andpistheprobabilityoftheeventAonanyoneBernoullitrial,thenIimP(-<)=forany£>0.(5.5.2)(频率具有稳定性)IfXn(n1)isindependent,withthenIimE<x)=(X)Sranx.xpopulation(总体)Definition6.2.1Apopulationisthesetofdataormeasurementsconsistsofallconceivablypossibleobservationsfromallobjectsinagivenphenomenon.Definition6.2.2Asampleisasubsetofthepopulationfromwhichpeoplecandrawconclusionsaboutthewhole.asamplefromthepopulationiscalledsampling.中位数IfarandomsamplehastheorderstatisticsX(“),then(i) TheSampleMedianisifnisoddrandomifniseven(ii) TheSampleRangeisHere,E(X)=isCanCdtheexpectationOfthemean.均值的期望-=iscalledthestandarderrorofthemean.均值的标准差X4n7.1PointEstimate点估计Definition7.1.1Supposeisaparameterofapopulation,X1,.,Xrtisarandomsamplefromthispopulation,andT(X,，X“)isastatisticthatisafunctionofXl,.Xn.Now,totheobservedvalue1,.,x,1,ifWeuseT(x1,.,xzj)asanestimatedvalueof0、thenT(X,，X)iscalledapointestimatorofandT(xp.,xzl)isreferredasapointestimateof.Thepointestimatorisalsooftenwrittenas.Unbiasedestimator(无偏估计量)Definition7.1.2.Supposeisanestimatorofaparameter.Thenisunbiasedifandonlyifminimumvarianceunbiasedestimator(最小方差无偏估计量)1.etbeanunbiasedestimatorof.Ifforany'whichisalsoanunbiasedestimatorof.wehaveD(O)D(O),theniscalledtheminimumvarianceunbiasedestimatorof.Sometimesitisalsocalledbestunbiasedestimator.3.MethodofMoments矩估计的方法Definition7.1.4SupposeXpX2,XconstitutearandomsamplefromthepopulationXthathaskunknownparametersv1k.Also,thepopulationhasfirskfinitemomentsE(X),E(X),E(X2)thatdependsontheunknownparameters.SolvethesystemofequationsE(X)=-X,.E(X2)=IJx,2'7，(7.1.4)E(Xk)=-XXlktogetunknownparametersexpressedbytheobservationsvalues,i.e.j=a(X,X?,Xk)forj=,2,k.Thenjisanestimatorofjbymethodofmoments.Supposethatisaparameterofapopulation,X1,.,Xnisarandomsampleoffromthispopulation,and=7(Xl,.,XJand2=(X1,.,Xzf)aretwostatisticssuchthatx<1,Ifforagivenawith0<<1,wehavePe1)=-a.ThenWereferto2asa(1-a)l00%confidenceintervalfor.Moreover,-aiscalledthedegreeofconfidence.。and2arecalledlowerandupperconfidencelimits.Theestimationusingconfidenceintervaliscalledintervalestimation.confidenceinterval置信区间lowerconfidencelimits置信下限upperconfidencelimits置信上限degreeofCOn行dence置信度2.极大似然函数likelihoodfunctionArandomsamplehastheobservedvaluesx1,x2,.,xrtfromapopulationwithanunknownparameter.Thenthelikelihoodfunctionforthissampleis1.()=f(xi,x2,.,xn)inwhich/(xpx2,.,xw;)isdefinedin(7.5.1).IycillllllLFllO±ATA5iil"l¾J41IlylJUIllSlS5dllclUlIlVilVl_WllJUClUiVconcerningoneormorepopulation.Astatisticalprocedureordecisionrulethatleadstoestablishingthetruthorfalsityofahypothesisiscalledastatisticaltest.显著性水平Definition8.2.1Whichaiscalledsignificantlevel,adescribeshowfarthesamplemeanisfarfromthepopulationmean.TwoTypesofErrorsDefinition8.2.3Ifithappensthatthehypothesisbeingtestedisactuallytrue,andiffromthesampleWereachtheconclusionthatitisfalse,WesaythatatypeIerrorhasbeencommitted.Ifithappensthathypothesisbeingtestedisactuallyfalse,andiffromthesamplewereachtheconclusionthatitistrue,wesaythatatypeIIerrorhasbeencommitted.