probability distributions什么是概率分布

Probability概率分 LearningObjectives學習目WhatisaProbabilityDistribution什么是概率分布Experiment,SampleSpace, 實驗,樣本空間,事RandomVariable,ProbabilityFunctions(pmf,pdf,cdf) 量,概率函DiscreteDistributions離散分Binomial 二項式分Poisson 泊松分布 超幾何分ContinuousDistributions連續(xù)分Normal 正態(tài)分Uniform 均勻分Exponential 指數(shù)分Logarithmicnormaldistribution對數(shù)正態(tài)分 威布爾分 WhatisaProbability什么是概率分布 rogressfromdescriptionofdatatowardsinferenceofdata,anconceptistheideaofaprobability當我們從描述性數(shù)據(jù)進步到推論性數(shù)據(jù)時,一個重要的內(nèi)容就是概率分布的念Toappreciatethenotionofaprobabilitydistribution,weneedtoreviewfundamentalconceptsrelatedto為了解概率分布的概念,我們需要復習各種基本相關概念Experiment,SampleSpace,實驗,樣本空間,事 量 WhatisaProbability什么是概率分布Anexperimentisanyactivitythatgeneratesasetofdata, aybenumericalor實驗是產(chǎn)生一系列數(shù)據(jù)的行為,數(shù)據(jù)有可能是數(shù)字的或非數(shù)字的

a

{1,2,..,

Experimentgeneratesnumerical/discrete實驗產(chǎn)生計數(shù)性實驗產(chǎn)生計數(shù)性實驗產(chǎn)生數(shù)字/離散數(shù)

Inspectingforstainmarks檢

實驗產(chǎn)生連實驗產(chǎn)生連續(xù)性

Experimentgeneratesattributedata

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ExperimentgeneratescontinuousdataWhatisaProbability什么是概率分布RandomExperiment隨機實Ifwethrowthediceagainandagain,orproducemanyshaftsfromthesameprocess,theeswillgenerallybedifferent,andcannotbepredictedinadvancewithtotalcertainty.如果我們擲子一次由一次,或從相同工序生產(chǎn)許多軸,結果會是不同的.不能完提前預測Anexperimentwhichcanresultindifferentes,eventhoughitisrepeatedthesamemannereverytime,iscalledarandom一個實驗導致不同的結果,即使它是每次以相同方式,這叫做隨機實 WhatisaProbability什么是概率分布SampleSpace樣本空Thecollectionofallpossible esofanexperimentiscalleditssamplespace.收集實驗的Tossingadice{1,2,3,…Inspectingforstain污點痕跡檢{PassFail}通過,不通{Allvaluesbetween0andmax.possible,say,10mm}所有值在0和最大值,可能的軸徑,例如 e,orasetof es,fromarandomexperimentiscalledanevent,i.e.itisasubsetofthesamplespace.一個結果,或一套結果,從一個隨機實驗出來的稱為事件,也就是樣本空間的子

WhatisaProbability什么是概率分布Example例1:Someeventsfromtossingofadice.從 的一些事件Event事件1: eisanoddnumber結果是奇 E1={1,3,Event事件2: eisanumber>4大于4的結 E2={5,Example2Someeventsfrommeasuringshaft?:從測量軸徑的一些事Eventeisdiametermean直徑大于平均Event事件eis E1={x>partfailingspecs.未通過規(guī)格的結果 E2={x<LSL,x> WhatisaProbability什么是概率分布RandomVariable Fromasameexperiment,differenteventscanbederiveddependingonaspectsoftheexperimentweconsider從一個相同的實驗,由于我們認為重要的實驗方面不同而產(chǎn)生不同的結Inmanycases,itisusefulandconvenienttodefinetheaspectoftheexperimentweareinterestedinbydenotingtheeventofinterestwithasymbol(usuallyanuppercaseletter),e.g.:許多方面,它是很有用和方便的定義我們 的實驗方面,通過一個大寫字母表示.舉例說明LetXbetheevent“thenumberofadiceis用X代表事件 的數(shù)字是奇數(shù)LetWbetheevent“theshaftiswithin用W代表事件”軸徑尺寸在規(guī)格 WhatisaProbability什么是概率分布 Wehavedefinedafunctionthatassignsarealnumbertoanexperimental ewithinthesamplespaceoftherandomexperiment.X=X=Partsoutofspecs.(LSL=8mm,USL=10mm)shaft

0..,7.99998,7.99999,8,9.99999,10,10.00001,10.00002,Thisfunction(XorWinourexamples)iscalledarandomvariable函數(shù)(例子中的X或W)稱為 esofthesameeventareclearlyuncertainandarevariablefrom eto ehasanequalchanceofbeing WhatisaProbability什么是概率分布 fyhowlikelya eofarandomvariablecanoccur,weassignanumericalvaluebetween0and1(or0to為量化一個 量的指定結果發(fā)生的可能性,我們指定一個數(shù)字介于0和1之間(Thisnumericalvalueiscalledtheprobabilityof Thereareafewwaysofinterpretingprobability.Acommonwayistointerpretprobabilityasafraction(orproportion)oftimesthe eoccursinmanyrepetitionsofthesamerandomexperiment.次Thismethodistherelativefrequencyapproachorfrequentistapproachtointerpreting這種方法概率解釋的相對頻率模擬或單位頻率模 WhatisaProbability什么是概率分布ProbabilityDistribution概率分Whenweareabletoassignaprobabilitytoeachpossible eofarandomvariableX,thefulldescriptionofalltheprobabilitiesassociatedwiththe esiscalledaprobabilitydistributionofX.當我們能夠表明一個 量的某一個可能結果的概率,則整個可能結果的概的描述稱為X的概率分Aprobabilitydistributionistypicallypresentedasacurveorplotthat一個概率分布被代表為一個曲線或點應有Allthepossible esofXonthehorizontalaxisTheprobabilityof eonthevertical每一個結果的概率在縱 關于概隨機現(xiàn) 隨機試 樣本點、樣本空

隨機事

事件的運算 對立、并、交、 WhatisaProbability什么是概率分布離散概率分布(理論上UniformUniformContinuousProbabilityDistributions 連續(xù)概率分布(理論上NormalNormal EmpiricalDistributions經(jīng)驗分Createdfromactualobservations.Usuallyrepresentedas根據(jù)實際觀測得來,通常用直方圖代Empiricaldistributions,liketheoreticaldistributions,applytobothdiscretecontinuous經(jīng)驗分布,象理論上的分布,適用于離散和連續(xù)分布 PropertiesofDistributions分布的描Threecommonimportantcharacteristics: definesnatureof形 definescentraltendencyof中 definesdispersionofdata(orDispersion,orScale) 統(tǒng)一分統(tǒng)一分

PropertiesofDistributions分布的描ShapeDescribeshowtheprobabilitiesofallthe esareCanbedescribedmathematicallywithanequationcalledaprobability可以用一個概率函數(shù)數(shù)字表示,舉例說1x

fx

e2

LowercaseletterrepresentsaspecificvalueofrandomvariableX小字母代表

f(x)meansP(X= PropertiesofDistributions分布的描Foradiscretedistribution,對于一個離散分布f(xcalledistheprobabilityf(x稱為概率集中:massfunction(pmf),e.g.:函數(shù),舉例說明=Foracontinuousdistribution,對于 f(x)iscalledtheprobabilityf(x)稱為概率密度densityfunction(pdf),e.g=ft

t

,tPPx xn xpx1 2120000 PropertiesofDistributions分布的描Thetotalprobabilityforanydistributionsumsto任何分布的全部概率總和為Inadiscretedistribution,probabilityisrepresentedasheightofthebar.在一個離散分布,概率用柱狀表

Inacontinuousdistribution,probabilityisrepresentedasareaunderthe(pdf),betweentwo 續(xù)分布,概率曲線下兩點間面積表

Normal PropertiesofDistributions分布的描ProbabilityofAnExactValueUnderPDFisForacontinuousrandomvariable,theprobabilityofanexactvalueoccurringistheoretically‘0’alineonapdfhas‘0’width, P(X=x)=

forarandomInpractice,ifweobtainaparticularvalue,e.g.12.57,ofarandomvariableX,howdoweinterprettheprobabilityof12.57happening? ItisinterpretedastheprobabilityofXassumingavaluewithinasmallintervalaround12.57,[12.565,,Thisisobtainedbyintegratingtheareaunderthepdfbetween12.565and和 PropertiesofDistributions分布的描Areaofalineisf(9.5)=P(X=9.5)=

Togetprobabilityof20.0,integrateareabetween19.995and20.005,i.e.P(19.995<X<Areadenotesprobabilityofgettingavaluebetween40.0and50.0. f(x)isusedtocalculateanthatrepresents注意:f(x)用于計算 概率的面 CumulativeDistributionFunction連續(xù)分布函Insteadofaprobabilitydistributionfunction,itisoftenusefultodescribe,foraspecificvaluexofarandomvariable,thetotalprobabilityofallpossiblevaluesoccurring,upto&includingx,i.e.P(Xx).代表一個概率分布函數(shù),它經(jīng)常用于描述, 量x的一個特定值,所有全可能發(fā)生的概率,包括xi.e.P(XAequationorfunctionthatlinksaspecificxvaluetothecumulatedprobabilitiesofallpossiblevaluesuptoandincludingxiscalledacumulativedistributionfunction(cdf),denotedasF(x).一個等式或函數(shù)相關于特定x值的累計F(x)=P(X

Comparef(x)=P(X= NormalDistribution正態(tài)分Normal ProbabilityDensity1x fx

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xCumulativeDistributionxxa

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fxdxx 1x e CumulativeDistributionFunction累計分布函ProbabilityProbabilityMassfx

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Theoreticallyderiveddistributionsusingcertainrandomexperimentsthatfrequentlyariseinapplications.Theoreticallyderiveddistributionsusingcertainrandomexperimentsthatfrequentlyariseinapplications.Usedtoesofsystemsthatbehavesimilarlyrandomexperimentsusedtoderivethedistributions.常用概率分DiscreteDistributions離散分 均勻分 二項式分 幾何分 超幾何分 泊松分Continuous 連續(xù)分 均勻分 正態(tài)分 指數(shù)分 威布爾分Erlang, 對數(shù)正態(tài)分 DiscreteDistributions重要的離散分Binomial Poisson Binomial BinomialDistributionBinomialExperimentAssumingwehaveaprocessthatishistoricallyknowntoproduceprejectpcanbeusedastheprobabilityoffindingafaileduniteachtimewedrawapartfromtheprocessforinspection.

Let’spullasampleofnpartsrandomlyfromalargepopulation(>10n)forinspection.讓我們隨機從一大批量樣本10n)中取出n個樣本Eachpartisclassifiedacceptor

SampleRejectRejectrate=BinomialDistributionBinomialExperimentAssumingwehaveaprocessthatishistoricallypcanbeusedastheoffindingafaileduniteach

Foreachtrial(drawingaunit),theofsuccessis對于每次試驗（取樣本），成概率是一個常wedrawapartfromtheprocessforinspection.P用于當我們從工序每次取出分時，取到不合格品的概率

Trialsareindependent;resultofaunitdoes eofnext試驗是獨立的，一個單位的結果不響下一個結果的輸出Let’spullasampleofnpartsrandomlyfromalargepopulation(>10n)forinspection.讓我們隨機從一大批量樣本10n)中取出n個樣本Eachpartisclassifiedacceptor

Eachtrialresultsinonlytwo 每一次試驗只有兩種可能的結果AAbinomial一個二項式試BinomialDistributionProbabilityMassFunction概率集中函Ifeachbinomialexperiment(pullingnpartsrandomlyforpass/failinspection)isseveraltimes,doweseethesamexdefectiveunitsallthe否可以每次看到相同的X不合格Thepmfthatdescribeshowthexdefectiveunits(calledsuccesses)aredistributedisn

,0Probabilityofgettingxdefectiveunits(xsuccesses)的概率（X合格品

xUsingasamplesizeofunits(n使用n個樣本量（n次

Giventhattheoveralldefectiveis(probabilityofsuccessis給出整個不合格品率(成功的概率是BinomialDistributionApplicationsThebinomialdistributionisextensivelyusedtomodelresultsofexperimentsthatgeneratees,e.g.pass/fail,go/nogo,accept/reject,二項式分布廣泛應用于結果只輸出兩種的實驗.舉例來說,通過/不通過,去/不去,接受 .等等Inindustrialpractice,itisusedfordatageneratedfromcountingofdefectives,在工業(yè)實際中,常用于缺陷品計數(shù)的數(shù)據(jù),Acceptance ProbabilityProbabilityofFindingX

Binomial

x

NumberofRejects BinomialDistributionExample1例Ifaprocesshistoricallygives10%rejectrate(p= 率是10%(p=0.10),whatisthechanceoffinding0,1,2or3defectiveswithinasampleof20units(n=20)?則對于20個樣本中發(fā)現(xiàn)0123缺陷品的概略是多少20

0

for

1

n xx BinomialDistributionExample1cont’d)例1繼Theseprobabilitiescanbeobtainedfrom這些概率可通過Minitab獲得CalcProbabilityDistributionsn=p=結果的指列 BinomialDistributionExample1From FromBinomial

n

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xProbabilityofProbabilityofFindingxDefectives

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Whatistheprobabilityofgetting2defectivesorless?BinomialDistributionExample1cont’d)例1(繼續(xù)Forthe2previouscharts,thex-axisdenotesthenumberofdefectiveunits,對于上頁中的圖表,X軸表明缺陷品單位的數(shù)量Ifwedivideeachxvaluebyconstantsamplesize,n,andre-expressthex-axisasaproportiondefectivep-axis,thedonot 代替X軸為缺陷品率p,則概率不變. BinomialDistribution二項式分ParametersofBinomialDistribution二項式分布的參Thelocation,dispersionandshapeofabinomialdistributionareaffectedbythesamplesize,n,anddefectiverate,p.

分布參 BinomialDistributionNormalApproximationtotheDependingonthevaluesofnandp,thebinomialdistributionsareafamilyofdistributionsthatcanbeskewedtotheleftorright.Undercertainconditions(combinationsofnandp),thebinomialdistribution approachestheshapeofanormal在一定的情況下(n和p一定),二項式分布近似于一個正態(tài)分布的形狀ForForp np>Forpfarfrom0.5(smalleror np> 2ppp1pnnnp1pBinomialDistributionMeanandVariance均值和方Althoughnandppindownaspecificbinomialdistribution,oftenthemeanvarianceofthedistributionareusedinpracticalapplicationssuchasthep-盡管n和p給定了一個特定的二項式分布,但分布的均值和方差經(jīng)常被實際的分布,象p-Themeanandvarianceofabinomial二項式分布的均值和方 DiscreteDistributions重要的離散分Binomial Poisson PoissonDistributionThisdistributionhavebeenfoundtoberelevantforapplicationsinvolvingerrorrates,particlecount,chemicalconcentration,etc,此分布被發(fā)現(xiàn)應用于錯誤率,灰塵數(shù),化學比,等等Px

e

xwhereisthemeannumberofevents(ordefectrate)withingivenunitoftimeor是給定的一個單位或空間中事件或缺陷率的平均數(shù)量And

SimeonD 2PoissonDistributionnumber esinatimeinterval(orspaceregion)isindependentof esinanotherinterval(orspaceprobabilityofanoccurrencewithi yshorttimeinterval(orspaceregion)isproportionaltothetimeinterval(orspaceregion)probabilityofmorethan1 eoccurringwithinashorttimeinterval(orspaceregion)isthemeanandvarianceforaPoissonDistribution PoissonDistributionThelocation,dispersionandshapeofaPoissondistributionisaffectedbythe泊松分布的位置,離散和形狀都受均影 Example2練習Acertainprocessyieldsadefectrateof4Foraopportunitiesinspected,determinetheprobability某一工序產(chǎn)生的缺陷率是4dpmo試計算其概 ExampleCalcProbabilityDistributionsa)ProbabilityMass b)CumulativeDistribution SummaryofApproximations近似總np>ifpnp>

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thesmallerthe&thelargerthetheif|p|

15Thelargerthe ContinuousDistributionsNormalExponential Normal正態(tài)分Normal NormalDistribution正態(tài)分Themostwidelyusedmodelforthedistributionofcontinuousrandom Arisesinthestudyofnumerousnaturalphysicalphenomena,suchasthevelocitymolecules,aswellasinoneofthemostimportantfindings,theCentralLimit來自于大量自然物理現(xiàn)象的研究,例如分子的電壓 NormalDistribution正態(tài)分Manynaturalphenomenaandman-madeprocessesareobservedtohavedistributions,orcanbecloselyrepresentedasnormally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,thelengthofamachinedpartisobservedtovaryaboutitsmeandue例如:機器元件的長度均值的變化由于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,轉速變化,Ifthesesourcesofvariationaresmall,independentandequallylikelytobepositiveornegativeaboutthemeanvalue,thelengthwillcloselyapproximateanormaldistribution. NormalDistribution正態(tài)分Normal ProbabilityDensity概率密度函1x fx

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X~N(,

分布ParametersofParametersofthe分布

Whatisthedifferencebetweenthe3normal三個正態(tài)分布有何不同SomePropertiesoftheNormal正態(tài)分布的一些

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Whatisthe betweenprocessA&foreachA,B分布的區(qū)別 SomePropertiesoftheNormal正態(tài)分布的一些Themean,medianandmodeallcoincideatthesamevalue.Thereis均值,中位數(shù)和重數(shù)一致為相同值完全對μ--

Doesitmeanthatanydatasetwhichhasmean,medianandmodeatthesamevaluewillautomaticallybeanormaldistribution?+SomePropertiesoftheNormal正態(tài)分布的一些Theareaundersectionsofthecurvecanbeusedtoestimatetheprobabilityofacertain“event”occurring:部分曲線下的面積可用于計算一定事件發(fā)生的概-

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+/-3isoftenreferredtoasthewidthofanormal+SomePropertiesoftheNormal正態(tài)分布的一些Let’scomputethecumulativeprobabilitiesofthefollowing讓我們計算下列分布的累計概

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P(X>20.0)P(X>20.0)=1–F(1.8)=P(XF(1.8)=P(X<

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- - +SomePropertiesoftheNormal正態(tài)分布的一些CalcProbabilityDistributions

x ZxSomePropertiesZx正態(tài)分布的一些Anormal