概率論與數(shù)理統(tǒng)計英文版總結(jié)資料概率論與數(shù)理統(tǒng)計英文版總結(jié)資料概率論與數(shù)理統(tǒng)計英文版總結(jié)資料合用標(biāo)準(zhǔn)SampleSpace樣本空間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)文檔合用標(biāo)準(zhǔn)Definition2.4.1EventsA1,A2,L,Anarecalledmutuallyexclusive,ifAiIAj,ij.IfAandBaremutuallyexclusive,thenP(AUB)P(A)P(B)(2.4.1)Mutuallyindependent事件的獨立性TwoeventsAandBaresaidtobeindependentifP(AIB)P(A)P(B)OrTwoeventsAandBareindependentifandonlyifP(B|A)P(B).ConditionalProbability條件概率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)文檔合用標(biāo)準(zhǔn)(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（貝葉斯公式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文檔合用標(biāo)準(zhǔn)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.文檔合用標(biāo)準(zhǔn)3.2DiscreteRandomVariables失散型隨機(jī)量DefinitionArandomvariableXiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequence{a1,a2,Lan,L}geometricdistribution（幾何分布）X1234?k?Ppq1q2q3qk－ppp1p?Binomialdistribution（二項分布）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文檔合用標(biāo)準(zhǔn)parameter,denotedbyP().Expectation(mean)數(shù)學(xué)希望DefinitionLetXbeadiscreterandomvariable.TheexpectationormeanofXisdefinedasE(X)xP(Xx)(3.3.1)x2．Variance方差standarddeviation（標(biāo)準(zhǔn)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概率密度函數(shù)DefinitionAfunctionf(x)definedon(,)iscalledaprobabilitydensityfunction（概率密度函數(shù)）if:f(x)0foranyxR;(ii)f(x)isintergrable(可積的)on(,)andf(x)dx1.文檔合用標(biāo)準(zhǔn)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.文檔合用標(biāo)準(zhǔn)Weeasilyget2D(X)x2f(x)dx2.(4.1.5).4.2UniformDistribution均勻分布Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunction1foraxb,f(x)ba0elsewhere,4.5ExponentialDistribution指數(shù)分布Definition4.5.1AcontinuousvariableXhasanexponentialdistributionwithparameter(0),ifitsdensityfunctionisgivenby1exforx0f(x)(4.5.1)0forx0ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenbyE(X),D(X)2.文檔合用標(biāo)準(zhǔn)4.3NormalDistribution正態(tài)分布1.DefinitionTheequationofthenormalprobabilitydensity,whosegraphisshowninFigure4.3.1,isf(x)1e(x)2/22x24.4NormalApproximationtotheBinomialDistribution（二項分布）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（聯(lián)合分布）Inthestudyofprobability,givenatleasttworandomvariablesX,Y,...,thataredefinedonaprobabilityspace,thejointprobabilitydistributionforX,Y,...isaprobability文檔合用標(biāo)準(zhǔn)distributionthatgivestheprobabilitythateachofX,Y,...fallsinanyparticularrangeordiscretesetofvaluesspecifiedforthatvariable.Conditionaldistribution條件分布ConsistentwiththedefinitionofconditionalprobabilityofeventswhenAistheeventX=xandBistheeventY=y,theconditionalprobabilitydistributionofXgivenY=yisdefinedaspX(x|y)p(x,y)forallxprovidedpY(y)

pY(y)0.Statisticalindependent隨機(jī)變量的獨立性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協(xié)方差和相關(guān)系數(shù)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).文檔合用標(biāo)準(zhǔn)Definition5.4.2TherandomvariablesXandYaresaidtobeuncorrelatediffCov(X,Y)0.中心5.5LawofLargeNumbersandCentralLimitTheorem極限制理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(頻率擁有牢固性)IfXn(n1)isindependent,withE(Xn),D(Xn)2,andSn*SnnnthenlimFn(x)(x)forallx.x文檔合用標(biāo)準(zhǔn)population(整體).Definition6.2.1Apopulationisthesetofdataormeasurementsconsistsofallconceivablypossibleobservationsfromallobjectsinagivenphenomenon.Apopulationmayconsistoffinitelyorinfinitelymanyvarieties.sample（樣本、子樣）Definition6.2.2Asampleisasubsetofthepopulationfromwhichsampling(抽樣)peoplecandrawconclusionsaboutthewhole.takingasample:Theprocessofperforminganexperimenttoobtainasamplefromthepopulationiscalledsampling.中位數(shù)DefinitionIfarandomsamplehastheorderstatisticsX(1),X(2),,X(n),then(i)TheSampleMedianisXn1)ifnisodd(M021XXnifnisevenn(1))((ii)TheSampleRangeisRX(n)X(1).SampleDistributions抽樣分布文檔合用標(biāo)準(zhǔn)1．samplingdistributionofthemean均值的抽樣分布Theorem6.3.1IfXismeanoftherandomsampleX1,X2,,XnofsizenfromarandomvariableXwhichhasmeanandthevariance2,then2E(X)andD(X).nItiscustomarytowriteE(X)asXandD(X)asHere,E(X)iscalledtheexpectationofthemean望iscalledthestandarderrorofthemean.Xn

2X..均值的期均值的標(biāo)準(zhǔn)差PointEstimate點估計DefinitionSupposeisaparameterofapopulation,X1,,Xnisarandomsamplefromthispopulation,andT(X1,,Xn)isastatisticthatisafunctionofX1,Xn.Now,totheobservedvaluex1,,xn,ifweuseT(x1,,xn)asanestimatedvalueof,thenT(X1,,Xn)iscalledapointestimatorofandT(x1,,xn)isreferredasapointestimateof.Thepointestimatorisalsooftenwrittenas?.Unbiasedestimator(無偏估計量)Definition7.1.2.Suppose?isanestimatorofaparameter.Then?isunbiasedifandonlyif文檔合用標(biāo)準(zhǔn)E(?).minimumvarianceunbiasedestimator（最小方差無偏估計量）Let?beanunbiasedestimatorof.Ifforany?'whichisalsoanunbiasedestimatorof,wehaveD(?)D(?'),then?iscalledtheminimumvarianceunbiasedestimatorof.Sometimesitisalsocalledbestunbiasedestimator.3.MethodofMoments矩估計的方法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.文檔合用標(biāo)準(zhǔn)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-----置信區(qū)間lowerconfidencelimits-----置信下限upperconfidencelimits-----置信上限deg