工程方法sigma概率分布

工程方法sigma概率分布LearningObjectives學(xué)習(xí)目得WhatisaProbabilityDistribution?什么就是概率分布?Experiment,SampleSpace,Event實驗,樣本空間,事件RandomVariable,ProbabilityFunctions(pmf,pdf,cdf)隨機變量,概率函數(shù)DiscreteDistributions離散分布BinomialDistribution二項式分布PoissonDistribution泊松分布、Hypergeometricdistribution超幾何分布ContinuousDistributions連續(xù)分布NormalDistribution正態(tài)分布Uniformdistribution均勻分布Exponentialdistribution指數(shù)分布Logarithmicnormaldistribution對數(shù)正態(tài)分布Weibulldistribution威布爾分布SamplingDistributions樣本分布ZDistributionZ分布tDistributiont分布c2Distributionc2

分布FDistributionF

分布2Asweprogressfromdescriptionofdatatowardsinferenceofdata,animportantconceptistheideaofaprobabilitydistribution、當(dāng)我們從描述性數(shù)據(jù)進(jìn)步到推論性數(shù)據(jù)時,一個重要得內(nèi)容就就是概率分布得概念、Toappreciatethenotionofaprobabilitydistribution,weneedtoreviewvariousfundamentalconceptsrelatedtoit:為了解概率分布得概念,我們需要復(fù)習(xí)各種基本相關(guān)概念:Experiment,SampleSpace,Event實驗,樣本空間,事件RandomVariable隨機變量、WhatisaProbabilityDistribution?

什么就是概率分布?Whatdowemeanbyinferenceofdata?3Experiment實驗Anexperimentisanyactivitythatgeneratesasetofdata,whichmaybenumericalornotnumerical、實驗就是產(chǎn)生一系列數(shù)據(jù)得行為,數(shù)據(jù)有可能就是數(shù)字得或非數(shù)字得、{1,2,、、,6}(a)Throwingadice擲骰子Experimentgeneratesnumerical/discretedataPinsStainsRejectAccept(b)Inspectingforstainmarks檢查污點印記ExperimentgeneratesattributedataPins(c)Measuringshaft?測量軸徑10.53mm10.49mm10.22mm10.29mm11.20mm……ExperimentgeneratescontinuousdataWhatisaProbabilityDistribution?

什么就是概率分布?實驗產(chǎn)生數(shù)字/離散數(shù)據(jù)實驗產(chǎn)生計數(shù)性數(shù)據(jù)實驗產(chǎn)生連續(xù)性數(shù)據(jù)4RandomExperiment隨機實驗Ifwethrowthediceagainandagain,orproducemanyshaftsfromthesameprocess,theouteswillgenerallybedifferent,andcannotbepredictedinadvancewithtotalcertainty、如果我們擲子一次由一次,或從相同工序生產(chǎn)許多軸,結(jié)果會就是不同得、不能完全提前預(yù)測、Anexperimentwhichcanresultindifferentoutes,eventhoughitisrepeatedinthesamemannereverytime,iscalledarandomexperiment、一個實驗導(dǎo)致不同得結(jié)果,即使她就是每次以相同方式,這叫做隨機實驗WhatisaProbabilityDistribution?

什么就是概率分布?5SampleSpace樣本空間Thecollectionofallpossibleoutesofanexperimentiscalleditssamplespace、收集實驗得所有可能結(jié)果稱為樣本空間Event事件Anoute,orasetofoutes,fromarandomexperimentiscalledanevent,i、e、itisasubsetofthesamplespace、一個結(jié)果,或一套結(jié)果,從一個隨機實驗出來得稱為事件,也就就是樣本空間得子集WhatisaProbabilityDistribution?

什么就是概率分布?6Event事件Example例1:Someeventsfromtossingofadice、從擲骰子得一些事件、Event事件1:theouteisanoddnumber結(jié)果就是奇數(shù)Event事件2:theouteisanumber>4大于4得結(jié)果Example例2:Someeventsfrommeasuringshaft?:從測量軸徑得一些事件Event事件1:theouteisadiameter>mean直徑大于平均值Event事件2:theouteisapartfailingspecs、未通過規(guī)格得結(jié)果、TE2={x<LSL,x>USL}TE2={5,6}TE1={1,3,5}TE1={x>m}WhatisaProbabilityDistribution?

什么就是概率分布?7RandomVariable隨機變量Fromasameexperiment,differenteventscanbederiveddependingonwhichaspectsoftheexperimentweconsiderimportant、從一個相同得實驗,由于我們認(rèn)為重要得實驗方面不同而產(chǎn)生不同得結(jié)果Inmanycases,itisusefulandconvenienttodefinetheaspectoftheexperimentweareinterestedinbydenotingtheeventofinterestwithasymbol(usuallyanuppercaseletter),e、g、:許多方面,她就是很有用和方便得定義我們感興趣得實驗方面,通過一個大寫得字母表示、舉例說明:LetXbetheevent“thenumberofadiceisodd”、用X代表事件”骰子得數(shù)字就是奇數(shù)”LetWbetheevent“theshaft?iswithinspecs、”、用W代表事件”軸徑尺寸在規(guī)格內(nèi)”WhatisaProbabilityDistribution?

什么就是概率分布?8RandomVariable隨機變量Wehavedefinedafunctionthatassignsarealnumbertoanexperimentaloutewithinthesamplespaceoftherandomexperiment、我們定義了一個函數(shù),其代表了一個在隨機實驗得樣本空間得一個真實實驗數(shù)字Thisfunction(XorWinourexamples)iscalledarandom

variablebecause:函數(shù)(例子中得X或W)稱為隨機變量,就是因為:Theoutesofthesameeventareclearlyuncertainandarevariablefromoneoutetoanother一個事件得發(fā)生結(jié)果就是明顯不定得,就是同另一個結(jié)果相異得、Eachoutehasanequalchanceofbeingselected、每一個結(jié)果有相同被選擇得機會、PinsMeasuringshaft?

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

什么就是概率分布?9大家有疑問的，可以詢問和交流可以互相討論下，但要小聲點Probability概率Toquantifyhowlikelyaparticularouteofarandomvariablecanoccur,wetypicallyassignanumericalvaluebetween0and1(or0to100%)、為量化一個隨機變量得指定結(jié)果發(fā)生得可能性,我們指定一個數(shù)字介于0和1之間(或0~100%)Thisnumericalvalueiscalledtheprobabilityoftheoute、這個數(shù)字稱為結(jié)果得概率Thereareafewwaysofinterpretingprobability、Amonwayistointerpretprobabilityasafraction

(orproportion)oftimestheouteoccursinmanyrepetitionsofthesamerandomexperiment、有幾種方式解釋概率、一般得方式就是解釋概率為在許多相同實驗重復(fù)后發(fā)生得分?jǐn)?shù)(或比例)次數(shù)Thismethodistherelativefrequencyapproachorfrequentistapproachtointerpretingprobability、這種方法概率解釋得相對頻率模擬或單位頻率模擬WhatisaProbabilityDistribution?

什么就是概率分布?11ProbabilityDistribution概率分布WhenweareabletoassignaprobabilitytoeachpossibleouteofarandomvariableX,thefulldescriptionofalltheprobabilitiesassociatedwiththepossibleoutesiscalledaprobabilitydistributionofX、當(dāng)我們能夠表明一個隨機變量得某一個可能結(jié)果得概率,則整個可能結(jié)果得概率得描述稱為X得概率分布Aprobabilitydistributionistypicallypresentedasacurveorplotthathas:一個概率分布被代表為一個曲線或點應(yīng)有:AllthepossibleoutesofXonthehorizontalaxisX得所有得可能結(jié)果在水平軸線上Theprobabilityofeachouteontheverticalaxis每一個結(jié)果得概率在縱軸上WhatisaProbabilityDistribution?

什么就是概率分布?12隨機現(xiàn)象隨機試驗樣本點、樣本空間語言表示事件得表示集合表示事件得特征

包含、相等隨機事件事件間得關(guān)系互斥事件得運算:對立、并、交、差

關(guān)于概率13NormalDistributionExponentialDistributionUniformDistributionBinomialDistributionDiscreteProbabilityDistributions(Theoretical)離散概率分布(理論上)ContinuousProbabilityDistributions(Theoretical)連續(xù)概率分布(理論上)WhatisaProbabilityDistribution?

什么就是概率分布?14EmpiricalDistributions經(jīng)驗分布Createdfromactualobservations、Usuallyrepresentedashistograms、根據(jù)實際觀測得來,通常用直方圖代表Empiricaldistributions,liketheoreticaldistributions,applytobothdiscreteandcontinuousdistributions、經(jīng)驗分布,象理論上得分布,適用于離散和連續(xù)分布、15Threemonimportantcharacteristics:三個常用重要Shape - definesnatureofdistribution形狀-定義分布得自然性Center - definescentraltendencyofdata中心-定義中心趨勢得數(shù)據(jù)Spread分布(或離散,或刻度) - definesdispersionofdata

(orDispersion,orScale)定義數(shù)據(jù)得離散PropertiesofDistributions分布得描述ExponentialDistributionUniformDistribution統(tǒng)一分布指數(shù)分布16Shape形狀Describeshowtheprobabilitiesofallthepossibleoutesaredistributed、描述所有可能結(jié)果可能性得分布Canbedescribedmathematicallywithanequationcalledaprobabilityfunction,e、g:可以用一個概率函數(shù)數(shù)字表示,舉例說明Probabilityfunction概率函數(shù)LowercaseletterrepresentsaspecificvalueofrandomvariableX小字母代表隨機變量X某一個特定值

f(x)

means

P(X=x)PropertiesofDistributions分布得描述1700f(t)1a2a3ab=4210、5ProbabilityFunctions概率函數(shù)Foradiscretedistribution,對于一個離散分布

f(x)calledistheprobabilityf(x)

稱為概率集中: massfunction(pmf),e、g、:函數(shù),舉例說明Foracontinuousdistribution,對于一個連續(xù)分布

f(x)iscalledtheprobabilityf(x)

稱為概率密度 densityfunction(pdf),

e、g、:函數(shù)舉例說明PropertiesofDistributions分布得描述18BinomialDistributionNormalDistributionThetotalprobabilityforanydistributionsumsto1、任何分布得全部概率總和為1Inadiscretedistribution, probabilityisrepresented asheightofthebar、在一個離散分布,概率用柱狀表示Inacontinuousdistribution, probabilityisrepresented asareaunderthecurve (pdf),betweentwopoints、在一個連續(xù)分布,概率用曲線下兩點間面積表示PropertiesofDistributions分布得描述19ProbabilityofAnExactValueUnderPDFisZero!PDF下一個準(zhǔn)確值得概率就是零Foracontinuousrandomvariable,theprobabilityofanexactvalueoccurringistheoretically‘0’becausealineonapdfhas‘0’width,implying:對于一個連續(xù)隨機變量,一個準(zhǔn)確值發(fā)生得概率理論上就是‘0,就是因為PDF上一條線得寬度就是‘0”、意味著:

Inpractice,ifweobtainaparticularvalue,e、g、12、57,ofarandomvariableX,howdoweinterprettheprobabilityof12、57happening?實際上,如果我們獲得一個特定得值,舉例說明、12、57,隨機變量X得一個值,我們?nèi)绾谓忉?2、57發(fā)生得概率、ItisinterpretedastheprobabilityofXassumingavaluewithinasmallintervalaround12、57,i、e、[12、565,12、575]、解釋為X假定一個值得概率在一個小間距在12、57左右,也就就是說[12、565,12、575]、Thisisobtainedbyintegratingtheareaunderthepdfbetween12、565and12、575、在PDF下12、565和12、575之間得整個面積為此點得概率、P(X=x)=0foracontinuousrandomvariablePropertiesofDistributions分布得描述20ExponentialDistributionAreaofalineiszero!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)用于計算一個代表概率得面積PropertiesofDistributions分布得描述21Insteadofaprobabilitydistributionfunction,itisoftenusefultodescribe,foraspecificvaluexofarandomvariable,thetotalprobabilityofallpossiblevaluesoccurring,upto&includingx,i、e、P(X

￡

x)、代表一個概率分布函數(shù),她經(jīng)常用于描述,隨機變量x得一個特定值,所有全部可能發(fā)生得概率,包括xi、e、P(X

￡

x)、Aequationorfunctionthatlinksaspecificxvaluetothecumulatedprobabilitiesofallpossiblevaluesuptoandincludingxiscalledacumulativedistributionfunction

(cdf),denotedasF(x)、一個等式或函數(shù)相關(guān)于特定x值得累計概率F(x)=P(X

x)CumulativeDistributionFunction連續(xù)分布函數(shù)pareagainst:f(x)

=

P(X=x)22CumulativeDistributionFunction累計分布函數(shù)NormalDistributionProbabilityDensityFunction概率密度函數(shù)NormalDistribution正態(tài)分布aa0.523ProbabilityMassFunctionCumulativeDistributionFunction累計分布函數(shù)CumulativeDistributionFunction累計分布函數(shù)24monProbabilityDistributions

常用概率分布DiscreteDistributions離散分布Uniform均勻分布Binomial二項式分布Geometric幾何分布Hypergeometric超幾何分布Poisson泊松分布ContinuousDistributions連續(xù)分布Uniform均勻分布Normal正態(tài)分布Exponential指數(shù)分布Weibull威布爾分布Erlang,Gamma？？？？？Lognormal對數(shù)正態(tài)分布Theoreticallyderiveddistributionsusingcertainrandomexperimentsthatfrequentlyariseinapplications、理論上,講分布就是隨機實驗大量應(yīng)用得來得Usedtomodeloutesofphysicalsystemsthatbehavesimilarlytorandomexperimentsusedtoderivethedistributions、通常自然系統(tǒng)模型輸出近似于隨機實驗25ImportantDiscreteDistributions重要得離散分布BinomialDistribution二項式分布PoissonDistribution泊松分布26BinomialDistribution二項式分布27BinomialExperiment二項式實驗Assumingwehaveaprocessthatishistoricallyknowntoproduceprejectrate、假設(shè)我們有一道工序,已知其歷史拒收率ppcanbeusedastheprobability offindingafaileduniteachtime wedrawapartfromtheprocess forinspection、P用于當(dāng)我們從工序每次取出一部分時,取到不合格品得概率。Let’spullasampleofnparts randomlyfromalargepopulation (>10n)forinspection、

讓我們隨機從一大批量樣本(>10n)中取出n個樣本Eachpartisclassifiedas acceptorreject、

每一部分被標(biāo)識接受或拒收。BinomialDistribution二項式分布Rejectrate=pSamplesize(n)28BinomialExperiment二項式實驗Assumingwehaveaprocessthatishistoricallyknowntoproduceprejectrate、假設(shè)我們有一道工序,已知其歷史拒收率ppcanbeusedastheprobability offindingafaileduniteachtime wedrawapartfromtheprocess forinspection、P用于當(dāng)我們從工序每次取出一部分時,取到不合格品得概率。Let’spullasampleofnparts randomlyfromalargepopulation (>10n)forinspection、

讓我們隨機從一大批量樣本(>10n)中取出n個樣本Eachpartisclassifiedas acceptorreject、

每一部分被標(biāo)識接受或拒收。BinomialDistribution二項式分布Foreachtrial(drawingaunit),theprobabilityofsuccessisconstant、對于每次試驗(取樣本),成功得概率就是一個常數(shù)Trialsareindependent;resultofaunitdoesnotinfluenceouteofnextunit試驗就是獨立得,一個單位得結(jié)果不影響下一個結(jié)果得輸出。Eachtrialresultsinonlytwopossibleoutes、每一次試驗只有兩種可能得結(jié)果。Abinomialexperiment!一個二項式試驗29ProbabilityMassFunction概率集中函數(shù)Ifeachbinomialexperiment(pullingnpartsrandomlyforpass/failinspection)isrepeatedseveraltimes,doweseethesamexdefectiveunitsallthetime?如果每一個二項式實驗(隨機取n個產(chǎn)品進(jìn)行通過/拒收檢查)被重復(fù)很多次,我們就是否可以每次看到相同得X不合格品Thepmfthatdescribeshowthexdefectiveunits(calledsuccesses)aredistributedisgivenas:PMF描述X個不合格品(也叫合格品)得如何分布,表示為Probabilityofgettingxdefectiveunits(xsuccesses)得到X不合格品品得概率(X合格品)Usingasamplesizeofnunits(ntrials)使用n個樣本量(n次)Giventhattheoveralldefectiverateisp(probabilityofsuccessisp)給出整個不合格品率p(成功得概率就是P)BinomialDistribution二項式分布30Applications應(yīng)用Thebinomialdistributionisextensivelyusedtomodelresultsofexperimentsthatgeneratebinaryoutes,e、g、pass/fail,go/nogo,accept/reject,etc、二項式分布廣泛應(yīng)用于結(jié)果只輸出兩種得實驗、舉例來說,通過/不通過,去/不去,接受/拒絕、等等、Inindustrialpractice,itisusedfordatageneratedfromcountingofdefectives,e、g、:在工業(yè)實際中,常用于缺陷品計數(shù)得數(shù)據(jù),舉例來說

1、AcceptanceSampling接受樣本 2、p-chartP-ChartBinomialDistribution二項式分布31Example1例1Ifaprocesshistoricallygives10%rejectrate(p=0、10),如果一個工序歷史上拒絕率就是10%(p=0、10),whatisthechanceoffinding0,1,2or3defectiveswithinasampleof20units(n=20)?則對于20個樣本中發(fā)現(xiàn)0,1,2或3缺陷品得概略就是多少?1、BinomialDistribution二項式分布32Example1(cont’d)例1繼續(xù)TheseprobabilitiescanbeobtainedfromMinitab:這些概率可通過Minitab獲得:

CalcaProbabilityDistributionsaBinomial…P(x)n=20p=0、1包含X個缺陷品得指定列存儲結(jié)果得指定列BinomialDistribution二項式分布33Example1(cont’d)FromExcel:FromMinitab:Whatistheprobabilityofgetting2defectivesorless?BinomialDistribution二項式分布34Example1(cont’d)例1(繼續(xù))Forthe2previouscharts,thex-axisdenotesthenumberofdefectiveunits,x、對于上頁中得圖表,X軸表明缺陷品單位得數(shù)量XIfwedivideeachxvalue byconstantsamplesize,n, andre-expressthex-axis asaproportiondefective

p-axis,theprobabilities donotchange、如果我們將X除以恒定得樣本量n,再重新代替X軸為缺陷品率p,則概率不變、BinomialDistribution二項式分布35Thelocation,dispersionandshapeofabinomialdistributionareaffectedbythesamplesize,n,anddefectiverate,p、二項式分布得位置,離散程度,和形狀受樣本量n和缺陷平率p影響、ParametersofBinomialDistribution二項式分布得參數(shù)分布參數(shù)BinomialDistribution二項式分布36NormalApproximationtotheBinomial二項式分布得正態(tài)近似Dependingonthevaluesofnandp,thebinomialdistributionsareafamilyofdistributionsthatcanbeskewedtotheleftorright、依靠不同得n和p,二項式分布就是一個傾斜至左邊或右邊得分布集合、Undercertainconditions(binationsofnandp),thebinomialdistributionapproximatelyapproachestheshapeofanormaldistribution:在一定得情況下(n和p一定),二項式分布近似于一個正態(tài)分布得形狀、Forp

?0、5, np>5Forpfarfrom0、5(smallerorlarger), np>10BinomialDistribution二項式分布37MeanandVariance均值和方差A(yù)lthoughnandppindownaspecificbinomialdistribution,oftenthemeanandvarianceofthedistributionareusedinpracticalapplicationssuchasthep-chart、盡管n

和p

給定了一個特定得二項式分布,但分布得均值和方差經(jīng)常被用于實際得分布,象p-chart、Themeanandvarianceofabinomialdistribution二項式分布得均值和方差orBinomialDistribution二項式分布38ImportantDiscreteDistributions重要得離散分布BinomialDistribution二項式分布PoissonDistribution泊松分布39PoissonDistribution泊松分布Thisdistributionhavebeenfoundtoberelevantforapplicationsinvolvingerrorrates,particlecount,chemicalconcentration,etc,此分布被發(fā)現(xiàn)應(yīng)用于錯誤率,灰塵數(shù),化學(xué)比,等等、where

isthemeannumberofevents(ordefectrate)withinagivenunitoftimeorspace、

就是給定得一個單位或空間中事件(或缺陷率)得平均數(shù)量、Andwhereissmall、40SimeonDPoisson41PoissonDistribution泊松分布Properties:numberofoutesinatimeinterval(orspaceregion)isindependentoftheoutesinanothertimeinterval(orspaceregion)單位時間(或空間)得數(shù)量輸出獨立于另一個單位時間(或空間)得數(shù)量輸出、probabilityofanoccurrencewithinaveryshorttimeinterval(orspaceregion)isproportionaltothetimeinterval(orspaceregion)在非常短時間(或空間)內(nèi)發(fā)生得概率就是單位時間(或單位空間)輸出數(shù)量得比率probabilityofmorethan1outeoccurringwithinashorttimeinterval(orspaceregion)isnegligible極短時間(空間單位)內(nèi)1個數(shù)量輸出得概率可忽略不記themeanandvarianceforaPoissonDistributionare泊松分布得均值和方差就是and42PoissonDistribution泊松分布Thelocation,dispersionandshapeofaPoissondistributionisaffectedbythemean、泊松分布得位置,離散和形狀都受均值影響43Example2練習(xí)2、Acertainprocessyieldsadefectrateof4dpmo、Foramillionopportunitiesinspected,determinetheprobabilitydistribution、某一工序產(chǎn)生得缺陷率就是4dpmo、試計算其概率分布、44Example2CalcProbabilityDistributionsPoissona)ProbabilityMassFunction b)CumulativeDistributionFunction45SummaryofApproximations近似總結(jié)Binomialp<0、1thesmallerthep&thelargerthenthebetter

15Thelargerthebetternp>5ifp?5np>10if|p|

?PoissonNormal46ImportantContinuousDistributionsNormalDistributionExponentialDistribution47NormalDistribution正態(tài)分布NormalDistribution48Themostwidelyusedmodelforthedistributionofcontinuousrandomvariables、連續(xù)性隨機變量應(yīng)用最廣泛得分布類型Arisesinthestudyofnumerousnaturalphysicalphenomena,suchasthevelocityofmolecules,aswellasinoneofthemostimportantfindings,theCentralLimitTheorem、來自于大量自然物理現(xiàn)象得研究,例如分子得電壓;中心極限定理也就是許多非常重要發(fā)現(xiàn)得其中之一、NormalDistribution正態(tài)分布49Manynaturalphenomenaandman-madeprocessesareobservedtohavenormaldistributions,orcanbecloselyrepresentedasnormallydistributed、我們觀測到許多自然現(xiàn)象和人為工序都符合正態(tài)分布,或近似于正態(tài)分布、Forexample,thelengthofamachinedpartisobservedtovaryaboutitsmeandueto:例如:機器元件得長度均值得變化由于:temperaturedrift,humiditychange,vibrations,cuttinganglevariations,cuttingtoolwear,bearingwear,rotationalspeedvariations,fixturingvariations,rawmaterialchangesandcontaminationlevelchanges溫度漂移,濕度變化,振動,切削角度變化,切削工具磨損,軸承磨損,轉(zhuǎn)速變化,夾具變化,原材料變更和污染級別變化,等等Ifthesesourcesofvariationaresmall,independentandequallylikelytobepositiveornegativeaboutthemeanvalue,thelengthwillcloselyapproximateanormaldistribution、如果上述來源變化較小,獨立和近似可能相對于均值偏正或偏負(fù),則長度近似于一個正態(tài)分布、NormalDistribution正態(tài)分布50CumulativeDistributionFunction累計分布函數(shù)NormalDistributionProbabilityDensityFunction概率密度函數(shù)NormalDistribution正態(tài)分布aa0.551Anormaldistributioncanbepletelydescribedbyknowingonlythe:一個正態(tài)分布完全可以描述由已知得Mean(m)均值Variance(s2)方差SomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性DistributionOneDistributionTwoDistributionThreeWhatisthedifferencebetweenthe3normaldistributions?三個正態(tài)分布有何不同?X~N(m,s2)1Parametersofthedistribution分布2分布3分布152WhatisthedifferencebetweenprocessA&Bforeachcase?A,B分布得區(qū)別?SomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性A~Normal(A,A2)B~Normal(B,B2)A~Normal(A,A2)B~Normal(B,B2)A~Normal(A,A2)B~Normal(B,B2)53Themean,medianandmodeallcoincideatthesamevalue-

m、Thereisperfectsymmetry、均值,中位數(shù)和重數(shù)一致為相同值-

m,完全對稱μ+￥-￥Doesitmeanthatanydatasetwhichhasmean,medianandmodeatthesamevaluewillautomaticallybeanormaldistribution?就是否上述三個參數(shù)一致得分布就就是正態(tài)分布?MeanMedianMode2SomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性54Theareaundersectionsofthecurvecanbeusedtoestimatetheprobabilityofacertain“event”occurring:部分曲線下得面積可用于計算一定事件發(fā)生得概率μPointofInflection1s+￥-￥68、27%95、45%99、73%+/-3sisoftenreferredtoasthewidthofanormaldistribution(常指正態(tài)分布得寬度)3SomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性55Let’sputethecumulativeprobabilitiesofthefollowingdistributions:讓我們計算下列分布得累計概率+￥-￥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

正態(tài)分布得一些特性56MiniTab:Calc

e

ProbabilityDistributions

e

Normal、、、EntermvalueEntersvalueEnterxvalueSomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性57Anormaldistributionwithm=0ands2=1iscalledastandardnormaldistribution、均值為0,方差為1得正態(tài)分布稱做標(biāo)準(zhǔn)正態(tài)分布、ObtainedthroughaZtransformation:通過Z轉(zhuǎn)化獲得、4SomePropertiesoftheNormalDistribution

正態(tài)分布得一些特性58Population總體Sample樣本SamplingDistributions抽樣分布59SamplingDistributions抽樣分布Whenwerepeatedlydrawsamplesofsizenfromagivenpopulationandputeasamplestatisticofinterest(`x,s,R,p,etc、),doweexpectthesamplestatistictobethesamevalueallthetime?當(dāng)我們重復(fù)從一個給定總體取出樣本量為n,計算想知道得樣本統(tǒng)計量,(`x,s,R,p,etc、),我們期望每次樣本統(tǒng)計量都相同?Theprobabilitydistributionofastatisticiscalledthesamplingdistributionofthestatistic、統(tǒng)計量得概率分布稱為統(tǒng)計量得抽樣分布E、g、,thedistributionof`Xiscalledthesamplingdistributionofthemean、舉例說明,X分布稱為均值得樣本分布Thesamplingdistributionofastatisticdependsonthedistributionofthepopulation,samplesizeandmethodofsampleselection、統(tǒng)計量得抽樣分布依靠于總體得分布,樣本量和抽樣方法、60ImportantSamplingDistributions重要得抽樣分布Z

Distributiont

Distribution2

DistributionFDistribution61ZDistribution分布62ZDistribution分布LetX1,X2,…,Xnbearandomsamplefromapopulationwithmeanmandstandarddeviations、ThestatistichasanormaldistributionwithThedistributionofZiscalledastandardnormaldistribution,denotedasZ~N(0,1)、ApplicationEstimating

basedon`xwhensisknownparing

vs

063ZDistribution分布讓X1,X2,…,Xn

就是從一個均值m和標(biāo)準(zhǔn)偏差s得總體中抽取得隨機樣本,則統(tǒng)計量就是就是這樣一個正態(tài)分布分布Z稱為標(biāo)準(zhǔn)正態(tài)分布,表示為Z~N(0,1)、應(yīng)用計算

根據(jù)`x當(dāng)s

已知

計算

對

064tDistribution分布65讓X1,X2,…,Xn

就是bearandomsamplefromanormalpopulationdistributionwithmeanmandstandarddeviations、Thestatistichasatdistributionwithn=n-1degreesoffreedom、InteractiveSlidetDistribution分布pareagainstZ,wehavereplacedsusingsintheTstatistic、66故有一個

t

分布其自由度

n=n-1InteractiveSlidetDistribution分布同Z比較,我們代替

s用

s

在

T

統(tǒng)計量讓X1,X2,…,Xn

就是從一個均值m和標(biāo)準(zhǔn)偏差s(未知)得總體中抽取得隨機樣本,則統(tǒng)計量就是67Thepreciseformofthetdistributiondependsonthedegreeofuncertaintyins2,whichisrepresentedbythedegreesoffreedomnfors2、T分布得精確形狀依靠于s2得不確定程度,其由S2得自由度來表示、Whennissmall,possibilityofmorevariationins2resultsingreaterprobabilityofextremedeviations,andhenceinaheaviertailedtdistribution、當(dāng)自由度較小,s2得更多變異導(dǎo)致最終偏差得可能變化,因此得到一個