工程方法概率分布介绍课件

ProbabilityDistributions

概率分布1ProbabilityDistributions

概率分布LearningObjectives学习目的WhatisaProbabilityDistribution?什么是概率分布?Experiment,SampleSpace,Event实验,样本空间,事件RandomVariable,ProbabilityFunctions(pmf,pdf,cdf)随机变量,概率函数DiscreteDistributions离散分布BinomialDistribution二项式分布PoissonDistribution泊松分布.Hypergeometricdistribution超几何分布ContinuousDistributions连续分布NormalDistribution正态分布Uniformdistribution均匀分布Exponentialdistribution指数分布Logarithmicnormaldistribution对数正态分布Weibulldistribution威布尔分布SamplingDistributions样本分布ZDistributionZ分布tDistributiont分布c2Distributionc2

分布FDistributionF

分布2LearningObjectives学习目的WhatisAsweprogressfromdescriptionofdatatowardsinferenceofdata,animportantconceptistheideaofaprobabilitydistribution.当我们从描述性数据进步到推论性数据时,一个重要的内容就是概率分布的概念.Toappreciatethenotionofaprobabilitydistribution,weneedtoreviewvariousfundamentalconceptsrelatedtoit:为了解概率分布的概念,我们需要复习各种基本相关概念:Experiment,SampleSpace,Event实验,样本空间,事件RandomVariable随机变量.WhatisaProbabilityDistribution?

什么是概率分布?Whatdowemeanbyinferenceofdata?3AsweprogressfromdescriptioExperiment实验Anexperimentisanyactivitythatgeneratesasetofdata,whichmaybenumericalornotnumerical.实验是产生一系列数据的行为,数据有可能是数字的或非数字的.{1,2,..,6}(a)Throwingadice掷骰子Experimentgeneratesnumerical/discretedataPinsStainsRejectAccept(b)Inspectingforstainmarks检查污点印记ExperimentgeneratesattributedataPins(c)MeasuringshaftÆ测量轴径10.53mm10.49mm10.22mm10.29mm11.20mm……ExperimentgeneratescontinuousdataWhatisaProbabilityDistribution?

什么是概率分布?实验产生数字/离散数据实验产生计数性数据实验产生连续性数据4Experiment实验{1,2,..,6}(a)ThRandomExperiment随机实验Ifwethrowthediceagainandagain,orproducemanyshaftsfromthesameprocess,theoutcomeswillgenerallybedifferent,andcannotbepredictedinadvancewithtotalcertainty.如果我们掷子一次由一次,或从相同工序生产许多轴,结果会是不同的.不能完全提前预测.Anexperimentwhichcanresultindifferentoutcomes,eventhoughitisrepeatedinthesamemannereverytime,iscalledarandomexperiment.一个实验导致不同的结果,即使它是每次以相同方式,这叫做随机实验WhatisaProbabilityDistribution?

什么是概率分布?5RandomExperiment随机实验WhatiSampleSpace样本空间Thecollectionofallpossibleoutcomesofanexperimentiscalleditssamplespace.收集实验的所有可能结果称为样本空间Event事件Anoutcome,orasetofoutcomes,fromarandomexperimentiscalledanevent,i.e.itisasubsetofthesamplespace.一个结果,或一套结果,从一个随机实验出来的称为事件,也就是样本空间的子集WhatisaProbabilityDistribution?

什么是概率分布?6SampleSpace样本空间WhatisaProbEvent事件Example例1:Someeventsfromtossingofadice.从掷骰子的一些事件.Event事件1:theoutcomeisanoddnumber结果是奇数Event事件2:theoutcomeisanumber>4大于4的结果Example例2:SomeeventsfrommeasuringshaftÆ:从测量轴径的一些事件Event事件1:theoutcomeisadiameter>mean直径大于平均值Event事件2:theoutcomeisapartfailingspecs.未通过规格的结果.ÞE2={x<LSL,x>USL}ÞE2={5,6}ÞE1={1,3,5}ÞE1={x>m}WhatisaProbabilityDistribution?

什么是概率分布?7Event事件ÞE2={x<LSL,x>RandomVariable随机变量Fromasameexperiment,differenteventscanbederiveddependingonwhichaspectsoftheexperimentweconsiderimportant.从一个相同的实验,由于我们认为重要的实验方面不同而产生不同的结果Inmanycases,itisusefulandconvenienttodefinetheaspectoftheexperimentweareinterestedinbydenotingtheeventofinterestwithasymbol(usuallyanuppercaseletter),e.g.:许多方面,它是很有用和方便的定义我们感兴趣的实验方面,通过一个大写的字母表示.举例说明:LetXbetheevent“thenumberofadiceisodd”.用X代表事件”骰子的数字是奇数”LetWbetheevent“theshaftÆiswithinspecs.”.用W代表事件”轴径尺寸在规格内”WhatisaProbabilityDistribution?

什么是概率分布?8RandomVariable随机变量WhatisaPRandomVariable随机变量Wehavedefinedafunctionthatassignsarealnumbertoanexperimentaloutcomewithinthesamplespaceoftherandomexperiment.我们定义了一个函数,其代表了一个在随机实验的样本空间的一个真实实验数字Thisfunction(XorWinourexamples)iscalledarandom

variablebecause:函数(例子中的X或W)称为随机变量,是因为:Theoutcomesofthesameeventareclearlyuncertainandarevariablefromoneoutcometoanother一个事件的发生结果是明显不定的,是同另一个结果相异的.Eachoutcomehasanequalchanceofbeingselected.每一个结果有相同被选择的机会.PinsMeasuringshaftÆ

X=Partsoutofspecs.(LSL=8mm,USL=10mm)0..,7.99998,7.99999,8,8,00001,…,9.99999,10,10.00001,10.00002,…LSLUSLWhatisaProbabilityDistribution?

什么是概率分布?9RandomVariable随机变量PinsMeasuriProbability概率Toquantifyhowlikelyaparticularoutcomeofarandomvariablecanoccur,wetypicallyassignanumericalvaluebetween0and1(or0to100%).为量化一个随机变量的指定结果发生的可能性,我们指定一个数字介于0和1之间(或0~100%)Thisnumericalvalueiscalledtheprobability

oftheoutcome.这个数字称为结果的概率Thereareafewwaysofinterpretingprobability.Acommonwayistointerpretprobabilityasafraction

(orproportion)oftimestheoutcomeoccursinmanyrepetitionsofthesamerandomexperiment.有几种方式解释概率.一般的方式是解释概率为在许多相同实验重复后发生的分数(或比例)次数Thismethodistherelativefrequencyapproachorfrequentistapproachtointerpretingprobability.这种方法概率解释的相对频率模拟或单位频率模拟WhatisaProbabilityDistribution?

什么是概率分布?10Probability概率WhatisaProbabiProbabilityDistribution概率分布WhenweareabletoassignaprobabilitytoeachpossibleoutcomeofarandomvariableX,thefulldescriptionofalltheprobabilitiesassociatedwiththepossibleoutcomesiscalledaprobabilitydistributionofX.当我们能够表明一个随机变量的某一个可能结果的概率,则整个可能结果的概率的描述称为X的概率分布Aprobabilitydistributionistypicallypresentedasacurveorplotthathas:一个概率分布被代表为一个曲线或点应有:AllthepossibleoutcomesofXonthehorizontalaxisX的所有的可能结果在水平轴线上Theprobabilityofeachoutcomeontheverticalaxis每一个结果的概率在纵轴上WhatisaProbabilityDistribution?

什么是概率分布?11ProbabilityDistribution概率分布Wh随机现象随机试验样本点、样本空间语言表示事件的表示集合表示事件的特征

包含、相等随机事件事件间的关系互斥事件的运算：对立、并、交、差

关于概率12随机现象随机试验样NormalDistributionExponentialDistributionUniformDistributionBinomialDistributionDiscreteProbabilityDistributions(Theoretical)离散概率分布(理论上)ContinuousProbabilityDistributions(Theoretical)连续概率分布(理论上)WhatisaProbabilityDistribution?

什么是概率分布?13NormalDistributionExponentialEmpiricalDistributions经验分布Createdfromactualobservations.Usuallyrepresentedashistograms.根据实际观测得来,通常用直方图代表Empiricaldistributions,liketheoreticaldistributions,applytobothdiscreteandcontinuousdistributions.经验分布,象理论上的分布,适用于离散和连续分布.14EmpiricalDistributions经验分布CreThreecommonimportantcharacteristics:三个常用重要Shape - definesnatureofdistribution形状-定义分布的自然性Center - definescentraltendencyofdata中心-定义中心趋势的数据Spread分布(或离散,或刻度) - definesdispersionofdata

(orDispersion,orScale)定义数据的离散PropertiesofDistributions分布的描述ExponentialDistributionUniformDistribution统一分布指数分布15ThreecommonimportantcharactShape形状Describeshowtheprobabilitiesofallthepossibleoutcomesaredistributed.描述所有可能结果可能性的分布Canbedescribedmathematicallywithanequationcalledaprobabilityfunction,e.g:可以用一个概率函数数字表示,举例说明Probabilityfunction概率函数LowercaseletterrepresentsaspecificvalueofrandomvariableX小字母代表随机变量X某一个特定值

f(x)

means

P(X=x)PropertiesofDistributions分布的描述16Shape形状ProbabilityfunctionLow00f(t)1a2a3ab=4210.5ProbabilityFunctions概率函数Foradiscretedistribution,对于一个离散分布

f(x)calledistheprobabilityf(x)

称为概率集中: massfunction(pmf),e.g.:函数,举例说明Foracontinuousdistribution,对于一个连续分布

f(x)iscalledtheprobabilityf(x)

称为概率密度 densityfunction(pdf),

e.g.:函数举例说明PropertiesofDistributions分布的描述1700f(t)1a2a3ab=4210.5ProbabilBinomialDistributionNormalDistributionThetotalprobabilityforanydistributionsumsto1.任何分布的全部概率总和为1Inadiscretedistribution, probabilityisrepresented asheightofthebar.在一个离散分布,概率用柱状表示Inacontinuousdistribution, probabilityisrepresented asareaunderthecurve (pdf),betweentwopoints.在一个连续分布,概率用曲线下两点间面积表示PropertiesofDistributions分布的描述18BinomialDistributionNormalDiProbabilityofAnExactValueUnderPDFisZero!PDF下一个准确值的概率是零Foracontinuousrandomvariable,theprobabilityofanexact

valueoccurringistheoretically‘0’becausealineonapdfhas‘0’width,implying:对于一个连续随机变量,一个准确值发生的概率理论上是‘0,是因为PDF上一条线的宽度是‘0”.意味着:

Inpractice,ifweobtainaparticularvalue,e.g.12.57,ofarandomvariableX,howdoweinterprettheprobabilityof12.57happening?实际上,如果我们获得一个特定的值,举例说明.12.57,随机变量X的一个值,我们如何解释12.57发生的概率.ItisinterpretedastheprobabilityofXassumingavaluewithinasmallintervalaround12.57,i.e.[12.565,12.575].解释为X假定一个值的概率在一个小间距在12.57左右,也就是说[12.565,12.575].Thisisobtainedbyintegratingtheareaunderthepdfbetween12.565and12.575.在PDF下12.565和12.575之间的整个面积为此点的概率.P(X=x)=0foracontinuousrandomvariablePropertiesofDistributions分布的描述19ProbabilityofAnExactValueExponentialDistributionAreaofalineiszero!f(9.5)=P(X=9.5)=0Togetprobabilityof20.0,integrateareabetween19.995and20.005,i.e.P(19.995<X<20.005)Areadenotesprobabilityofgettingavaluebetween40.0and50.0.Note:

f(x)isusedtocalculateanareathatrepresentsprobability注意:f(x)用于计算一个代表概率的面积PropertiesofDistributions分布的描述20ExponentialDistributionAreaoInsteadofaprobabilitydistributionfunction,itisoftenusefultodescribe,foraspecificvaluexofarandomvariable,thetotalprobabilityofallpossiblevaluesoccurring,upto&includingx,i.e.P(X

£

x).代表一个概率分布函数,它经常用于描述,随机变量x的一个特定值,所有全部可能发生的概率,包括xi.e.P(X

£

x).Aequationorfunctionthatlinksaspecificxvaluetothecumulatedprobabilitiesofallpossiblevaluesuptoandincludingxiscalledacumulativedistributionfunction

(cdf),denotedasF(x).一个等式或函数相关于特定x值的累计概率F(x)=P(Xx)CumulativeDistributionFunction连续分布函数Compareagainst:f(x)

=

P(X=x)21InsteadofaprobabilitydistrCumulativeDistributionFunction累计分布函数NormalDistributionProbabilityDensityFunction概率密度函数NormalDistribution正态分布aa0.522CumulativeDistributionFunctiProbabilityMassFunctionCumulativeDistributionFunction累计分布函数CumulativeDistributionFunction累计分布函数23ProbabilityMassFunctionCumulCommonProbabilityDistributions

常用概率分布DiscreteDistributions离散分布Uniform均匀分布Binomial二项式分布Geometric几何分布Hypergeometric超几何分布Poisson泊松分布ContinuousDistributions连续分布Uniform均匀分布Normal正态分布Exponential指数分布Weibull威布尔分布Erlang,Gamma？？？？？Lognormal对数正态分布Theoreticallyderiveddistributionsusingcertainrandomexperimentsthatfrequentlyariseinapplications.理论上,讲分布是随机实验大量应用得来的Usedtomodeloutcomesofphysicalsystemsthatbehavesimilarlytorandomexperimentsusedtoderivethedistributions.通常自然系统模型输出近似于随机实验24CommonProbabilityDistributioImportantDiscreteDistributions重要的离散分布BinomialDistribution二项式分布PoissonDistribution泊松分布25ImportantBinomialDistributionBinomialDistribution二项式分布26BinomialDistribution26BinomialExperiment二项式实验Assumingwehaveaprocessthatishistoricallyknowntoproduceprejectrate.假设我们有一道工序，已知其历史拒收率pp

canbeusedastheprobability

offindingafaileduniteachtime wedrawapartfromtheprocess forinspection.P用于当我们从工序每次取出一部分时，取到不合格品的概率。Let’spullasampleofnparts randomlyfromalargepopulation (>10n)forinspection.

让我们随机从一大批量样本(>10n)中取出n个样本Eachpartisclassifiedas acceptorreject.

每一部分被标识接受或拒收。BinomialDistribution二项式分布Rejectrate=pSamplesize(n)27BinomialExperiment二项式实验BinomiBinomialExperiment二项式实验Assumingwehaveaprocessthatishistoricallyknowntoproduceprejectrate.假设我们有一道工序，已知其历史拒收率pp

canbeusedastheprobability

offindingafaileduniteachtime wedrawapartfromtheprocess forinspection.P用于当我们从工序每次取出一部分时，取到不合格品的概率。Let’spullasampleofnparts randomlyfromalargepopulation (>10n)forinspection.

让我们随机从一大批量样本(>10n)中取出n个样本Eachpartisclassifiedas acceptorreject.

每一部分被标识接受或拒收。BinomialDistribution二项式分布Foreachtrial(drawingaunit),theprobabilityofsuccessisconstant.对于每次试验（取样本），成功的概率是一个常数Trialsareindependent;resultofaunitdoesnotinfluenceoutcomeofnextunit试验是独立的，一个单位的结果不影响下一个结果的输出。Eachtrialresultsinonlytwopossibleoutcomes.每一次试验只有两种可能的结果。Abinomialexperiment!一个二项式试验28BinomialExperiment二项式实验BinomiProbabilityMassFunction概率集中函数Ifeachbinomialexperiment(pullingnpartsrandomlyforpass/failinspection)isrepeatedseveraltimes,doweseethesamexdefectiveunitsallthetime?如果每一个二项式实验（随机取n个产品进行通过/拒收检查）被重复很多次，我们是否可以每次看到相同的X不合格品Thepmfthatdescribeshowthe

xdefectiveunits(calledsuccesses)aredistributedisgivenas:PMF描述X个不合格品（也叫合格品）的如何分布，表示为Probabilityofgettingxdefectiveunits(xsuccesses)得到X不合格品品的概率（X合格品）Usingasamplesizeofnunits(ntrials)使用n个样本量（n次）Giventhattheoveralldefectiverateisp(probabilityofsuccessisp)给出整个不合格品率p(成功的概率是P）BinomialDistribution二项式分布29ProbabilityMassFunction概率集中函Applications应用Thebinomialdistributionisextensivelyusedtomodelresultsofexperimentsthatgeneratebinaryoutcomes,e.g.pass/fail,go/nogo,accept/reject,etc.二项式分布广泛应用于结果只输出两种的实验.举例来说,通过/不通过,去/不去,接受/拒绝.等等.Inindustrialpractice,itisusedfordatageneratedfromcountingofdefectives,e.g.:在工业实际中,常用于缺陷品计数的数据,举例来说

1.AcceptanceSampling接受样本 2.p-chartP-ChartBinomialDistribution二项式分布30Applications应用BinomialDistribExample1例1Ifaprocesshistoricallygives10%rejectrate(p=0.10),如果一个工序历史上拒绝率是10%(p=0.10),whatisthechanceoffinding0,1,2or3defectiveswithinasampleof20units(n=20)?则对于20个样本中发现0,1,2或3缺陷品的概略是多少?1.BinomialDistribution二项式分布31Example1例1BinomialDistributiExample1(cont’d)例1继续TheseprobabilitiescanbeobtainedfromMinitab:这些概率可通过Minitab获得:

CalcâProbabilityDistributionsâBinomial…P(x)n=20p=0.1包含X个缺陷品的指定列存储结果的指定列BinomialDistribution二项式分布32Example1(cont’d)例1继续P(x)n=Example1(cont’d)FromExcel:FromMinitab:Whatistheprobabilityofgetting2defectivesorless?BinomialDistribution二项式分布33Example1(cont’d)FromExcel:FExample1(cont’d)例1(继续)Forthe2previouscharts,thex-axisdenotesthenumberofdefectiveunits,x.对于上页中的图表,X轴表明缺陷品单位的数量XIfwedivideeach

xvalue byconstantsamplesize,n, andre-expressthex-axis asaproportiondefective

p-axis,theprobabilities donotchange.如果我们将X除以恒定的样本量n,再重新代替X轴为缺陷品率p,则概率不变.BinomialDistribution二项式分布34Example1(cont’d)例1(继续)BinomiThe

location,dispersion

and

shape

ofabinomialdistributionareaffectedbythe

samplesize,

n,

and

defectiverate,p.二项式分布的位置,离散程度,和形状受样本量n和缺陷平率p影响.ParametersofBinomialDistribution二项式分布的参数分布参数BinomialDistribution二项式分布35Thelocation,dispersionandsNormalApproximationtotheBinomial二项式分布的正态近似Dependingonthevaluesofnandp,thebinomialdistributionsareafamilyofdistributionsthatcanbeskewedtotheleftorright.依靠不同的n和p,二项式分布是一个倾斜至左边或右边的分布集合.Undercertainconditions(combinationsofnandp),thebinomialdistributionapproximatelyapproachestheshapeofanormaldistribution:在一定的情况下(n和p一定),二项式分布近似于一个正态分布的形状.For

p

»0.5, np>5Forpfarfrom0.5(smallerorlarger), np>10BinomialDistribution二项式分布36NormalApproximationtotheBiMeanandVariance均值和方差Althoughnandppindownaspecificbinomialdistribution,oftenthemeanandvarianceofthedistributionareusedinpracticalapplicationssuchasthep-chart.尽管n

和p

给定了一个特定的二项式分布,但分布的均值和方差经常被用于实际的分布,象p-chart.Themeanandvarianceofabinomialdistribution二项式分布的均值和方差orBinomialDistribution二项式分布37MeanandVariance均值和方差orBinomImportantDiscreteDistributions重要的离散分布BinomialDistribution二项式分布PoissonDistribution泊松分布38ImportantBinomialDistributionPoissonDistribution泊松分布Thisdistributionhavebeenfoundtoberelevantforapplicationsinvolvingerrorrates,particlecount,chemicalconcentration,etc,此分布被发现应用于错误率,灰尘数,化学比,等等.whereisthemeannumberofevents(ordefectrate)withinagivenunitoftimeorspace.是给定的一个单位或空间中事件(或缺陷率)的平均数量.Andwhereissmall.39PoissonDistribution泊松分布ThisdSimeonDPoisson40SimeonDPoisson40PoissonDistribution泊松分布Properties:numberofoutcomesinatimeinterval(orspaceregion)isindependentoftheoutcomesinanothertimeinterval(orspaceregion)单位时间(或空间)的数量输出独立于另一个单位时间(或空间)的数量输出.probabilityofanoccurrencewithinaveryshorttimeinterval(orspaceregion)isproportionaltothetimeinterval(orspaceregion)在非常短时间(或空间)内发生的概率是单位时间(或单位空间)输出数量的比率probabilityofmorethan1outcomeoccurringwithinashorttimeinterval(orspaceregion)isnegligible极短时间(空间单位)内1个数量输出的概率可忽略不记themeanandvarianceforaPoissonDistributionare泊松分布的均值和方差是and41PoissonDistribution泊松分布ProperPoissonDistribution泊松分布Thelocation,dispersionandshapeofaPoissondistributionisaffectedbythemean.泊松分布的位置,离散和形状都受均值影响42PoissonDistribution泊松分布TheloExample2练习2.Acertainprocessyieldsadefectrateof4dpmo.Foramillionopportunitiesinspected,determinetheprobabilitydistribution.某一工序产生的缺陷率是4dpmo.试计算其概率分布.43Example2练习2.AcertainprocessExample2CalcProbabilityDistributionsPoissona)ProbabilityMassFunction b)CumulativeDistributionFunction44Example2CalcProbabilityDiSummaryofApproximations近似总结Binomialp<0.1thesmallerthep&thelargerthenthebetter15Thelargerthebetternp>5ifp»5np>10if|p|

½Poisson

Normal

45SummaryofApproximations近似总结BImportantContinuousDistributionsNormalDistributionExponentialDistribution46ImportantNormalDistribution46NormalDistribution正态分布NormalDistribution47NormalDistributionNormalDistThemostwidelyusedmodelforthedistributionofcontinuousrandomvariables.连续性随机变量应用最广泛的分布类型Arisesinthestudyofnumerousnaturalphysicalphenomena,suchasthevelocityofmolecules,aswellasinoneofthemostimportantfindings,theCentralLimitTheorem.来自于大量自然物理现象的研究,例如分子的电压;中心极限定理也是许多非常重要发现的其中之一.NormalDistribution正态分布48ThemostwidelyusedmodelforManynaturalphenomenaandman-madeprocessesareobservedtohavenormaldistributions,orcanbecloselyrepresentedasnormallydistributed.我们观测到许多自然现象和人为工序都符合正态分布,或近似于正态分布.Forexample,thelengthofamachinedpartisobservedtovaryaboutitsmeandueto:例如:机器元件的长度均值的变化由于:temperaturedrift,humiditychange,vibrations,cuttinganglevariations,cuttingtoolwear,bearingwear,rotationalspeedvariations,fixturingvariations,rawmaterialchangesandcontaminationlevelchanges温度漂移,湿度变化,振动,切削角度变化,切削工具磨损,轴承磨损,转速变化,夹具变化,原材料变更和污染级别变化,等等Ifthesesourcesofvariationaresmall,independentandequallylikelytobepositiveornegativeaboutthemeanvalue,thelengthwillcloselyapproximateanormaldistribution.如果上述来源变化较小,独立和近似可能相对于均值偏正或偏负,则长度近似于一个正态分布.NormalDistribution正态分布49ManynaturalphenomenaandmanCumulativeDistributionFunction累计分布函数NormalDistributionProbabilityDensityFunction概率密度函数NormalDistribution正态分布aa0.550CumulativeDistributionFunctiAnormaldistributioncanbecompletelydescribedbyknowingonlythe:一个正态分布完全可以描述由已知的Mean(m)均值Variance(s2)方差SomePropertiesoftheNormalDistribution

正态分布的一些特性DistributionOneDistributionTwoDistributionThreeWhatisthedifferencebetweenthe3normaldistributions?三个正态分布有何不同?X~N(m,s2)1Parametersofthedistribution分布2分布3分布151AnormaldistributioncanbecWhatisthedifferencebetweenprocessA&Bforeachcase?A,B分布的区别?SomePropertiesoftheNormalDistribution

正态分布的一些特性A~Normal(A,A²)B~Normal(B,B²)A~Normal(A,A²)B~Normal(B,B²)A~Normal(A,A²)B~Normal(B,B²)52WhatisthedifferenceSomePrThemean,medianandmodeallcoincideatthesamevalue-

m.Thereisperfectsymmetry.均值,中位数和重数一致为相同值-

m,完全对称µ+¥-¥Doesitmeanthatanydatasetwhichhasmean,medianandmodeatthesamevaluewillautomaticallybeanormaldistribution?是否上述三个参数一致的分布就是正态分布?MeanMedianMode2SomePropertiesoftheNormalDistribution

正态分布的一些特性53Themean,medianandmodeallTheareaundersectionsofthecurvecanbeusedtoestimatetheprobabilityofacertain“event”occurring:部分曲线下的面积可用于计算一定事件发生的概率µPointofInflection1s+¥-¥68.27%95.45%99.73%+/-3sisoftenreferredtoasthewidthofanormaldistribution(常指正态分布的宽度)3SomePropertiesoftheNormalDistribution

正态分布的一些特性54TheareaundersectionsoftheLet’scomputethecumulativeprobabilitiesofthefollowingdistributions:让我们计算下列分布的累计概率+¥-¥m=3.5s=0.61.8+¥-¥20.0m=16.6s=2.8+¥-¥m=-1.5s=0.9-2.80.5F(1.8)=P(X<1.8)??P(X>20.0)=1–F(20.0)??P(-2.8<X<0.5)=??(a)(b)(c)SomePropertiesoftheNormalDistribution

正态分布的一些特性55Let’scomputethecumulativepMiniTab:Calc

ð

ProbabilityDistributions

ð

Normal...EntermvalueEntersvalueEnterxvalueSomePropertiesoftheNormalDistribution

正态分布的一些特性56MiniTab:EnterEnterEnterSomeAnormaldistributionwithm=0ands2=1iscalledastandardnormaldistribution.均值为0,方差为1的正态分布称做标准正态分布.ObtainedthroughaZtransformation:通过Z转化获得.4SomePropertiesoftheNormalDistribution

正态分布的一些特性57Anormaldistributionwithm=Population总体Sample样本SamplingDistributions抽样分布58Population总体Sample样本SamplingDSamplingDistributions抽样分布Whenwerepeatedlydrawsamplesofsize

nfromagivenpopulationandcomputeasamplestatisticofinterest(`x,s,R,p,etc.),doweexpectthesamplestatistictobethesamevalueallthetime?当我们重复从一个给定总体取出样本量为n,计算想知道的样本统计量,(`x,s,R,p,etc.),我们期望每次样本统计量都相同?Theprobabilitydistributionofastatisticiscalledthesamplingdistributionofthestatistic.统计量的概率分布称为统计量的抽样分布E.g.,thedistributionof`Xiscalledthesamplingdistributionofthemean.举例说明,X分布称为均值的样本分布Thesamplingdistributionofastatisticdependsonthedistributionofthepopulation,samplesizeandmethodofsampleselection.统计量的抽样分布依靠于总体的分布,样本量和抽样方法.59SamplingDistributions抽样分布WhenImportantSamplingDistributions重要的抽样分布Z

Distributiont

Distribution²

DistributionFDistribution60ImportantZDistribution60ZDistribution分布61ZDistribution分布61ZDistribution分布LetX1,X2,…,Xnbearandomsamplefromapopulationwithmeanmandstandarddeviations.ThestatistichasanormaldistributionwithThedistributionofZiscalledastandardnormaldistribution,denotedasZ~N(0,1).ApplicationEstimatingbasedon`xwhensisknownComparingvs062ZDistribution分布LetX1,X2,…ZDistribution分布让X1,X2,…,Xn

是从一个均值m和标准偏差s的总体中抽取的随机样本,则统计量是是这样一个正态分布分布Z称为标准正态分布,表示为Z~N(0,1).应用计算根据`x当s

已知

计算对063ZDistribution分布让X1,X2,…,tDistribution分布64tDistribution分布64让X1,X2,…,Xn

是bearandomsamplefromanormalpopulationdistributionwithmeanmandstandarddeviations.Thestatistichasatdistributionwithn=n-1degreesoffreedom.InteractiveSlidetDistribution分布CompareagainstZ,wehavereplacedsusingsintheTstatistic.65让X1,X2,…,Xn是bearandom故有一个

t

分布其自由度

n=n-1InteractiveSlidetDistribution分布同Z比较,我们代替

s用

s

在

T

统计量让X1,X2,…,Xn

是从一个均值m和标准偏差s(未知)的总体中抽取的随机样本,则统计量是66故有一个t分布其自由度n=n-1InteractThepreciseformofthetdistributiondependsonthedegreeofuncertaintyins2,whichisrepresentedbythedegreesoffreedomnfors2.T分布的精确形状依靠于s2的不确定程度,其由S2的自由度来表示.Whennissmall,possibilityofmorevariationins2resultsingreaterprobabilityofextremedeviations,andhenceinaheaviertailedtdistribution.当自由度较小,s2的更多变异导致最终偏差的可能变化,因此得到一个大尾巴的t分布Whenntendstowardsinfinity,thereisnouncertaintyintheestimates2,andthetdistributionbecomesthestandardnormaldistribution.当自由度趋向于无穷大,计算值S2无不定度,T分布成为标准正态分布.tDistribution分布ApplicationEstimatingbasedon`xwhensisunknownComparingvs0应用计算根据`x当s

已知

计算对067ThepreciseformofthetdistProbabilityFunctions概率函数BinomialDistribution二项式分布PDF下一个准确值的概率是零Event事件1:theoutcomeisanoddnumber结果是奇数ExperimentgeneratesattributedataRandomVariable随机变量用W和Y分别代表自由度为u和n的独立的c2分布,则其比率为:RandomVariable随机变量Toquantifyhowlikelyaparticularoutcomeofarandomvariablecanoccur,wetypicallyassignanumericalvaluebetween0and1(or0to100%).CentralLimitTheorem中心极限定理Aequationorfunctionthatlinksaspecificxvaluetothecumulatedprobabilitiesofallpossiblevaluesuptoandincludingxiscalledacumulativedistributionfunction(cdf),denotedasF(x).asareaunderthecurveAbinomialexperiment!betweenprocessA&BInteractiveSlide²Distribution卡方分布68ProbabilityFunctions概率函数²DiLetX1,X2,…,Xnbearandomsamplefromanormalpopulationdistributionwithm