《风险评价教学》PPT课件.ppt

Widely usedDistributionsinRiskWhatisthePoissonDistribution?WhatistheWeibullDistribution?风险评价基础(第二讲)Simeon Denis Poisson n“Researches on the probability of criminal and civil verdicts”1837（犯罪和民法裁决）.nlooked at the form of the binomial distribution when the number of trials was large（试验的次数较大时）.nHe derived the cumulative Poisson distribution as the limiting case of the binomial when the chance of success tend to zero（成功的机会趋于0）.PoissonDistributionPOISSON(x,mean,cumulative)nXisthenumberofevents.nMeanistheexpectednumericvalue.nCumulativeisalogicalvaluethatdeterminestheformoftheprobabilitydistributionreturned.IfcumulativeisTRUE,POISSONreturnsthecumulativePoissonprobabilitythatthenumberofrandomeventsoccurringwillbebetweenzeroandxinclusive;ifFALSE,itreturnsthePoissonprobabilitymassfunctionthatthenumberofeventsoccurringwillbeexactlyx.PoissonandbinomialDistributionDefinitionsA binomial probability distribution results from a procedure that meets all the following requirements:1.The procedure has a fixed number of trials.2.The trials must be independent.(The outcome of any individual trial doesnt affect the probabilities in the other trials.)3.Each trial must have all outcomes classified into two categories.4.The probabilities must remain constant for each trial.Notation for Binomial Probability DistributionsS and F(success and failure)denote two possible categories of all outcomes;p and q will denote the probabilities of S and F,respectively,soP(S)=p(p=probability of success)P(F)=1 p=q(q=probability of failure)Notation(cont)n denotes the number of fixed trials.x denotes a specific number of successes in n trials,so x can be any whole number between 0 and n,inclusive.p denotes the probability of success in one of the n trials.q denotes the probability of failure in one of the n trials.P(x)denotes the probability of getting exactly x successes among the n trials.Important Hintsv Be sure that x and p both refer to the same category being called a success.v When sampling without replacement,the events can be treated as if they were independent if the sample size is no more than 5%of the population size.(That is n is less than or equal to 0.05N.)Methods for Finding ProbabilitiesWe will now present three methods for finding the probabilities corresponding to the random variable x in a binomial distribution.Method 1:Using the Binomial Probability Formula P(x)=px qn-x(n x)!x!n!for x=0,1,2,.,nwheren=number of trialsx=number of successes among n trialsp=probability of success in any one trialq=probability of failure in any one trial(q=1 p)Method 2:UsingTable A-1 in Appendix APart of Table A-1 is shown below.With n=4 and p=0.2 in the binomial distribution,the probabilities of 0,1,2,3,and 4 successes are 0.410,0.410,0.154,0.026,and 0.002 respectively.Poisson and binominal Distribution Poisson&binominalDistributionnAsalimittobinomialwhennislargeandpissmall.nAtheorembySimeonDenisPoisson(1781-1840).Parameterl=np=expectedvaluenAsnislargeandpissmall,thebinomialprobabilitycanbeapproximatedbythePoissonprobabilityfunctionnP(X=x)=e-llx/x!,wheree=2.71828nIonchannelmodeling:n=numberofchannelsincellsandpisprobabilityofopeningforeachchannel;BinomialandPoissonapproximationAdvantage:Noneedtoknownandpestimatetheparameterl fromdata200 yearly reports of death by horse-kick from10 cavalry corps over a period of 20 years in 19th century by Prussian officials(骑兵部队).Pool the last two cells and conduct a chi-square test to see if Poisson model is compatible with data or not.Degree of freedom is 4-1-1=2.Pearsons statistic=.304;P-value is.859(you can only tell it is between.95 and.2 from table in the book);accept null hypothesis,data compatible with model RutherfoldandGeiger(1910)卢瑟福和盖革nPolonium(钚)sourceplacedashortdistancefromasmallscreen.Foreachof2608eighth-minuteintervals,theyrecordedthenumberofalphaparticlesimpingingonthescreenMedical Imaging:X-ray,PET scan(positron emission tomography),MRI(Magnetic Resonance Imaging )(核磁共振检查)Other related application inPoisson process for modeling number of event occurrences in a spatial(空间的)or temporal domain(时间的区域)Homogeneity(同一性):rate of occurrence is uniformIndependent occurrence in non-overlapping areas(非叠加)PoissonDistributionnAdiscreteRVX followsthePoissondistributionwithparameterlifitsprobabilitymassfunctionis:nWideapplicabilityinmodelingthenumberofrandomeventsthatoccurduringagiventimeintervalThe Poisson Process:nCustomersthatarriveatapostofficeduringadaynWrongphonecallsreceivedduringaweeknStudentsthatgototheinstructorsofficeduringofficehoursnandpacketsthatarriveatanetworkswitchPoissonDistribution(cont.)nMeanandVariancenProof:SumofPoissonRandomVariablesnXi,i=1,2,n,areindependentRVsnXifollowsPoissondistributionwithparameterlinPartialsumdefinedas:nSnfollowsPoissondistributionwithparameterlPoissonApproximationtoBinomialnBinomialdistributionwithparameters(n,p)nAsnandp0,withnp=lmoderate,binomialdistributionconvergestoPoissonwithparameterlnProof:ModelingArrivalStatisticsnPoissonprocesswidelyusedtomodelpacketarrivalsinnumerousnetworkingproblemsnJustification:providesagoodmodelforaggregatetrafficofalargenumberof“independent”usersJMostimportantreasonforPoissonassumption:Analytictractability（分析处理）ofqueueingmodels（排队模型）。POISSONDISTRIBUTIONn例题：如果电话号码本中每页的错误个数为2.3个，K为每页中错误数目的随机变量。（a）画出它的概率密度和累积分布图；（b）求足以满概括50%页数中差错误的K。n根据公式：n可以求出等的概率。关于概率分布曲线以及累计概率分布曲线的绘制和分析的问题:(1)离散分布；(2)其代表的具体意义。n例题：某单位每月发生事故的情况如下：n每月的事故数012345n频数（月数）27128210n注意：一共是50个月的统计资料：n根据如上的数据，认为n（a）最有可能的是每月发生一次事故，这正确吗？n（b）在均值上下各的范围是多少？n（a）解：每月发生一次事故概率为：nn（b）在均值上下各的范围是多少？应用泊松分布解题的步骤如下：应用泊松分布解题的步骤如下：检查前提假设是否成立。最主要的条件是在每一标准单位内所指的事件发生的概率是常数；泊松分布用来泊松分布用来计算标准单位（一张照片、一只机翼、一块材料等等）计算标准单位（一张照片、一只机翼、一块材料等等）内的缺陷数、交通死亡人数等等，在排队理论中占有内的缺陷数、交通死亡人数等等，在排队理论中占有重要的地位。重要的地位。n确定变量，求出值；n求对应个别K的泊松分布概率；n求若干个K的泊松分布概率的总和；n求泊松分布的均值和方差；n画出概率分布和累积分布图。Dr.WallodiWeibullnThe Weibull distribution is by far the worldsmostpopularstatisticalmodelforlifedata（寿命数据）.It is also used in many otherapplications,suchasweatherforecastingandfittingdataofallkinds（数据拟合）.Amongallstatistical techniques it may be employed forengineeringanalysiswithsmallersamplesizesthananyothermethod.Havingresearchedandappliedthismethodforalmosthalfacentury。nWaloddiWeibullwasbornonJune18,1887.His family originally came fromSchleswig-Holstein,atthattimecloselyconnectedwithDenmark.Therewereanumberoffamousscientistsandhistoriansinthefamily.Hisowncareerasanengineerandscientistiscertainlyanunusualone.nHewasamidshipmanintheRoyalSwedishCoast Guard in 1904 was promoted tosublieutenantin1907,Captainin1916,andMajorin1940.HetookcoursesattheRoyalInstitute of Technology where he laterbecameafullprofessor(1924)andgraduatedin1924.HisdoctorateisfromtheUniversityofUppsalain1932.HeworkedinSwedish and German industries as aninventor(ball and roller bearings,electrichammer,)andasaconsultingengineer.MyfriendsatSAABinTrollhattenSwedengavemesomeofWeibullspapers.SAABisoneofmanycompaniesthatemployedWeibullasaconsultant.BackgroundBackgroundWWaloddi Weibull(1887-1979)aloddi Weibull(1887-1979)invented theinvented the Weibull Weibull distribution in1937.distribution in1937.His 1951 paper represents the culmination(His 1951 paper represents the culmination(顶顶峰峰 )of his work in reliability analysis.of his work in reliability analysis.The U.S.Air Force recognized the merit of Weibulls The U.S.Air Force recognized the merit of Weibullsmethods and funded his research to 1975.methods and funded his research to 1975.Leonard Johnson at Genral Motors,improved Weibulls Leonard Johnson at Genral Motors,improved Weibulls methods.Weibull used mean rank values for plottingmethods.Weibull used mean rank values for plottingbut Johnson suggested the use of median rank values.but Johnson suggested the use of median rank values.nHis first paper was on the propagation ofexplosive wave in 1914.He took part inexpeditionstotheMediterranean,theCaribbean,and the Pacific ocean on theresearchship“Albatross”wherehedevelopedthe technique of using explosive charges todeterminethetypeofoceanbottomsedimentsandtheirthickness,justaswedotodayinoffshoreoilexploration（地震波技术来测量沉积岩的种类和厚度）。nHe published many papers on strength ofmaterials,fatigue,ruptureinsolids,bearings,and of course,the Weibull distribution.Theauthorhasidentified65paperstodateplushis excellent book on fatigue analysis(1),1961.27ofthesepaperswerereportstotheUS Air Force at Wright Field on Weibullanalysis.(MostofthesereportstoWPAFBareno longer available even from NTIS.Theauthor would appreciate copies of WeibullspapersfromtheWPAFBfiles.)Dr.WeibullwasafrequentvisitortoWPAFB.nHismostfamouspaper(2)presentedintheUSA,was given before the ASME in 1951,using seven case studies with Weibulldistributions.Many,including the author,were skeptical that this method of allowingthe data to select the most appropriatedistributionfromthebroadfamilyofWeibulldistributionswouldwork.Howevertheearlysuccess of the method with very smallsamplesatPratt&WhitneyAircraftcouldnotbe ignored.Further,Dorian Shainin,aconsultant for Pratt&Whitney,stronglyencouragedtheuseofWeibullanalysis.Theauthorsoonbecameabeliever.nRobert Heller(3)spoke at the 1984SymposiumtotheMemoryofWaloddiWeibullinStockholm,Swedenandsaid,n“In 1963,at the invitation of the ProfessorFreudenthal,hebecameaVisitingProfessorat Columbia Universitys Institute for theStudyofFatigueandReliability.IwaswiththeInstituteatthattimeandgottoknowDr.Weibull personally.I learned a great dealfrom him and from Emil Gumbel and fromFreudenthal,thethreefoundersofProbabilistic Mechanics of Structures andMaterials.It was interesting to watch thefriendlyrivalrybetweenGumbel,thetheoretician and the two engineers,WeibullandFreudenthal.”n“The Extreme Value family of distributions,to which both the Gumbel and the Weibulltypebelong,ismostapplicabletomaterials,structures and biologicalsystems because ithas an increasing failure rate and candescribewearoutprocesses.Well,thesetwomen,bothintheirlateseventiesatthetime,showedthatthesedistributionsdidnotapplytothem.Theydidnotwearoutbutwerefulloflifeandenergy.Gumbelwentskiingeveryweekend and when I took Dr.and Mrs.WeibulltotheRooseveltHomeinHydeParkonacoldwinterday,herefusedmyofferedarmtohelphimontheicywalkwayssaying:“A little ice and snow never bothered aSwede.”nIn 1941 BOFORS,a Swedish armsfactory,gave him a personal researchprofessorshipinTechnicalPhysicsattheRoyalInstituteofTechnology,Stockholm.nIn1972,theAmericanSocietyofMechanicalEngineers(4)awardedDr.WeibulltheirgoldmedalcitingProfessorWeibullas“apioneerinthestudyoffracture,fatigue,andreliabilitywhohascontributedtotheliteratureforoverthirty years.His statistical treatment ofstrength and life has found widespreadapplicationinengineeringdesign.”Theawardwas presented by Dr.Richard Folsom,PresidentofASME,andPresidentofRensselaer Polytechnic Institute when theauthor was a student there.By coincidencethe author received the 1988 ASME goldmedal for statistical contributions includingadvancementsinWeibullanalysis.nThe author has an unconfirmed storytoldbyfriendsatWrightPattersonAirForce Base that Dr.Weibull was in agreatstateofhappinessonhislastvisittolectureattheAirForceInstitute ofTechnologyin1975ashehadjustbeenmarriedtoaprettyyoungSwedishgirl.He was 88 years old at the time.Hisfirstwifehaspassedonearlier.Itwason this trip that the photo above wastaken at the University of Washingtonwherehealsolectured.nTheUSAirForceMaterialsLaboratoryshouldbe commended for encouraging WaloddiWeibullformanyyearswithresearchcontracts.The author is also indebted toWPAFB for contracting the original USAFWeibull Analysis Handbook(5)and Weibullvideo training tape,as hewastheprincipalauthor of both.The latest version of thatHandbook is the fourth edition of The NewWeibullHandbook(6).nProfessorWeibullsproudestmomentcamein1978whenhereceivedtheGreatGoldmedalfromtheRoyalSwedishAcademyofEngineering Sciences,which was personallypresentedtohimbyKingCarlXVIGustavofSwedennHewasdevotedtohisfamilyandwasproudofhisninechildrenandnumerousgrandandgreat-grandchildren.nDr.Weibull was a member of manytechnical societies and worked to thelastdayofhisremarkablelife.HediedonOctober12,1979inAnnecy,France.nTheWeibullDistributionwasfirstpublishedin1939,over60yearsagoandhasproventobe invaluable for life data analysis inaerospace,automotive,electricpower,nuclear power,medical,dental,electronics,every industry.Yettheauthor isfrustratedthatonlythreeuniversitiesintheUSAteachWeibull analysis.To encourage the use ofWeibull analysis the author provides freecopies of The New Weibull Handbook touniversitylibrariesinEnglishspeakingcountriesthatrequestthebook.ThecorrespondingSuperSMITHsoftwareisavailable from Wes Fulton in demo versionfreefromhisWebsite.n()BackgroundBackground proved that the Weibull distribution and the smallest extreame value distributions(Type III)are same.The engineers at Pratt&Whitney found that the Weibull method worked well with extremely small samples,even 2 or 3 failures.Advantages of Weibull AnalysisAdvantages of Weibull Analysis Small SamplesSmall SamplesThe primary advantage of Weibull analysis is the ability to The primary advantage of Weibull analysis is the ability to provide failure analysis and failure forecasts accurately with provide failure analysis and failure forecasts accurately with small samples.Furthermore,small samples also allow cost small samples.Furthermore,small samples also allow cost Effective component testing.Effective component testing.Graphical AnalysisGraphical AnalysisAnother advantage of Weibull analysis is that it have a simple Another advantage of Weibull analysis is that it have a simple and useful graphical plot.It can be easily generated with and useful graphical plot.It can be easily generated with cumulative probability paper.cumulative probability paper.Advantages of Weibull AnalysisAdvantages of Weibull Analysis Application AreasApplication Areas Failure forecasting and prediction,Failure forecasting and prediction,Evaluating corrective action plans,Evaluating corrective action plans,Engineering change substantiation,Engineering change substantiation,Maintenance planning and cost effective Maintenance planning and cost effective replacement strategies,replacement strategies,Spare parts forecasting,Spare parts forecasting,Warranty analysis and support cost predictions,Warranty analysis and support cost predictions,Example:Example:In a certain project,In a certain project,“How many failures will we have in the next six month or“How many failures will we have in the next six month or a year?”a year?”To do a scheduled maintenance or prepare spares,To do a scheduled maintenance or prepare spares,“How many units will be needed for doing overhauling in“How many units will be needed for doing overhauling in the near future?”the near future?”After an engineering change,After an engineering change,“How many units must be tested for verifying that the old“How many units must be tested for verifying that the old failure mode is eliminated or improved with confidence failure mode is eliminated or improved with confidence level?”level?”Weibull