概率论与数理统计英文版总结

Sample?Space??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,denotedby,isalsoanevent,calledanimpossibleevent,meansthatitneveroccursintheexperiment.Probabilityofevents（概率）Ifthenumberofsuccessesinntrailsisdenotedbys,andifthesequenceofrelativefrequencies/snobtainedforlargerandlargervalueofnapproachesalimit,thenthislimitisdefinedastheprobabilityofsuccessinasingletrial.“equallylikelytooccur”------probability（古典概率）IfasamplespaceSconsistsofNsamplepoints,eachisequallylikelytooccur.AssumethattheeventAconsistsofnsamplepoints,thentheprobabilitypthatAoccursis()npPANMutuallyexclusive(互斥事件)Definition2.4.1Events12,,,nAAAarecalledmutuallyexclusive,if,ijAAij.Theorem2.4.1IfAandBaremutuallyexclusive,then()()()PABPAPB(2.4.1)Mutuallyindependent事件的独立性TwoeventsAandBaresaidtobeindependentif()()()PABPAPBOrTwoeventsAandBareindependentifandonlyif(|)()PBAPB.ConditionalProbability条件概率Theprobabilityofaneventisfrequentlyinfluencedbyotherevents.DefinitionTheconditionalprobabilityofB,givenA,denotedby(|)PBA,isdefinedby()(|)()PABPBAPAif()0PA.(2.5.1)Themultiplicationtheorem乘法定理If12k,,,AAAareevents,then12k121312121()()(|)(|)(|)kkPAAAPAPAAPAAAPAAAAIftheevents12k,,,AAAareindependent,thenforanysubset12{,,,}{1,2,,}miiik,1212()()()()mmPAAAPAPAPAiiiiii(全概率公式totalprobability)Theorem2.6.1.Iftheevents12,,,kBBBconstituteapartitionofthesamplespaceSsuchthat()0jPBfor1,2,,,jkthanforanyeventAofS,11()()()()kkjjjjjPAPABPBPAB(2.6.2)（贝叶斯公式Bayes’formula.）Theorem2.6.2Iftheevents12,,,kBBBconstituteapartitionofthesamplespaceSsuchthat()0jPBfor1,2,,,jkthanforanyeventAofS,()0PA,1()(|)(|)()(|)iiikjjjPBPABPBAPBPAB.for1,2,,ik(2.6.2)ProofBythedefinitionofconditionalprobability,()(|)()iiPBAPBAPAUsingthetheoremoftotalprobability,wehave1()(|)(|)()(|)iiikjjjPBPABPBAPBPAB1,2,,ik1.randomvariabledefinitionDefinition3.1.1Arandomvariableisarealvaluedfunctiondefinedonasamplespace;i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.2.DistributionfunctionDefinition3.1.2LetXbearandomvariableonthesamplespaceS.Thenthefunction()()FXPXx.RxiscalledthedistributionfunctionofXNoteThedistributionfunction()FXisdefinedonrealnumbers,notonsamplespace.3.PropertiesThedistributionfunction()FxofarandomvariableXhasthefollowingproperties:(1)()Fxisnon-decreasing.Infact,if12xx,thentheevent1{}Xxisasubsetoftheevent2{}Xx,thus1122()()()()FxPXxPXxFx(2)()lim()0xFFx,()lim()1xFFx.(3)ForanyRx0,0000lim()(0)()xxFxFxFx.Thisistosay,thedistributionfunction()FxofarandomvariableXisrightcontinuous.3.2DiscreteRandomVariables离散型随机变量Definition3.2.1ArandomvariableXiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequence12{,,,}naaageometricdistribution（几何分布）X1234…k…Ppq1pq2pq3pqk－1p…Binomialdistribution（二项分布）Definition3.4.1ThenumberXofsuccessesinnBernoullitrialsiscalledabinomialrandomvariable.Theprobabilitydistributionofthisdiscreterandomvariableiscalledthebinomialdistributionwithparametersnandp,denotedby(,)Bnp.poissondistribution（泊松分布）Definition3.5.1AdiscreterandomvariableXiscalledaPoissonrandomvariable,ifittakesvaluesfromtheset{0,1,2,},andif()(;)!kPXkpkek,00,1,2,k(3.5.1)Distribution(3.5.1)iscalledthePoissondistributionwithparameter,denotedby()P.Expectation(mean)数学期望Definition3.3.1LetXbeadiscreterandomvariable.TheexpectationormeanofXisdefinedas()()xEXxPXx(3.3.1)2．Variance方差standarddeviation（标准差）Definition3.3.2LetXbeadiscreterandomvariable,havingexpectation()EX.ThenthevarianceofX,denoteby()DXisdefinedastheexpectationoftherandomvariable2()X2()()DXEX(3.3.6)Thesquarerootofthevariance()DX,denoteby()DX,iscalledthestandarddeviationofX:122()DXEX(3.3.7)probabilitydensityfunction概率密度函数Definition4.1.1Afunctionf(x)definedon(,)iscalledaprobabilitydensityfunction（概率密度函数）if:(i)()0foranyfxxR;(ii)f(x)isintergrable(可积的)on(,)and()1fxdx.Definition4.1.2Letf(x)beaprobabilitydensityfunction.IfXisarandomvariablehavingdistributionfunction()()()xFxPXxftdt,(4.1.1)thenXiscalledacontinuousrandomvariablehavingdensityfunctionf(x).Inthiscase,2112()()xxPxXxftdt.(4.1.2)5.Mean（均值）6.variance方差Similarly,thevarianceandstandarddeviationofacontinuousrandomvariableXisdefinedby22()(())DXEX,(4.1.4)Where()EXisthemeanofX,isreferredtoasthestandarddeviation.Weeasilyget222()()DXxfxdx.(4.1.5).4.2UniformDistribution均匀分布Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunction1for,()0elsewhere,axbfxbaDefinition4.1.2LetXbeacontinuousrandomvariablehavingprobabilitydensityfunctionf(x).Thenthemean(orexpectation)ofXisdefinedby()()EXxfxdx,(4.1.3)providedtheintegralconvergesabsolutely.4.5ExponentialDistribution指数分布Theorem4.5.1ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenby2(),()EX