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)0forx0ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistrib