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1、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 d
2、erived 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.nCumulativeisalogicalvaluethatdeterminestheformoftheprobabilitydistribut
3、ionreturned.IfcumulativeisTRUE,POISSONreturnsthecumulativePoissonprobabilitythatthenumberofrandomeventsoccurringwillbebetweenzeroandxinclusive;ifFALSE,itreturnsthePoissonprobabilitymassfunctionthatthenumberofeventsoccurringwillbeexactlyx.PoissonandbinomialDistributionDefinitionsA binomial probabilit
4、y 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
5、 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 suc
6、cess)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
7、 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
8、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 Probab
9、ility 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. W
10、ith 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(1781-1
11、840).Parameterl=np=expectedvaluenAsnislargeandpissmall,thebinomialprobabilitycanbeapproximatedbythePoissonprobabilityfunctionnP(X=x)=e-llx/x!,wheree=2.71828nIonchannelmodeling:n=numberofchannelsincellsandpisprobabilityofopeningforeachchannel;BinomialandPoissonapproximationxn=100,p=.01Poisson0.366032
12、.3678791.36973.3678792.184865.1839403.06099.0613134.014942.0153285.002898.0030666.0000463.0005117Advantage:Noneedtoknownandpestimatetheparameterl fromdataX=Numberofdeathsfrequencies01091652223341total200200 yearly reports of death by horse-kick from10 cavalry corps over a period of 20 years in 19th
13、century by Prussian officials(骑兵部队).xDatafrequenciesPoissonprobabilityExpectedfrequencies0109.5435108.7165.331566.3222.10120.233.02054.141.0030.6200Pool 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 s
14、tatistic = .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,theyrecordedthenumberofal
15、phaparticlesimpingingonthescreenMedical Imaging : X-ray, PET scan (positron emission tomography), MRI (Magnetic Resonance Imaging ) (核磁共振检查)Other related application in#ofaparticlesObservedfreq.Expectedfreq.057541203211238340735255264532508540839462732547139140845689272910101111+66Poisson process fo
16、r 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:nWideappli
17、cabilityinmodelingthenumberofrandomeventsthatoccurduringagiventimeintervalThe Poisson Process:nCustomersthatarriveatapostofficeduringadaynWrongphonecallsreceivedduringaweeknStudentsthatgototheinstructorsofficeduringofficehoursnandpacketsthatarriveatanetworkswitchPoissonDistribution(cont.)nMeanandVar
18、iancenProof:SumofPoissonRandomVariablesnXi , i =1,2,n,areindependentRVsnXifollowsPoissondistributionwithparameterlinPartialsumdefinedas:nSnfollowsPoissondistributionwithparameterlPoissonApproximationtoBinomialnBinomialdistributionwithparameters(n,p)nAsnandp0,withnp=lmoderate,binomialdistributionconv
19、ergestoPoissonwithparameterlnProof:ModelingArrivalStatisticsnPoissonprocesswidelyusedtomodelpacketarrivalsinnumerousnetworkingproblemsnJustification:providesagoodmodelforaggregatetrafficofalargenumberof“independent”usersJMostimportantreasonforPoissonassumption:Analytictractability（分析处理）ofqueueingmod
20、els（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。 012345678910 1 1 2 6 24 120 720 5040 40320 362880362888000.10030.23060.26520.20330.11690.05380.02060.00680.00190.00050.0001 0.1003 0.3309 0.5961 0.7994 0.9163 0.9701 0.9907 0.
21、9975 0.9994 0.9999 1.0000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
22、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
23、 used in many otherapplications,suchasweatherforecastingandfittingdataofallkinds（数据拟合）.Amongallstatistical techniques it may be employed forengineeringanalysiswithsmallersamplesizesthananyothermethod.Havingresearchedandappliedthismethodforalmosthalfacentury。nWaloddiWeibullwasbornonJune18,1887. His f
24、amily originally came fromSchleswig-Holstein,atthattimecloselyconnectedwithDenmark.Therewereanumberoffamousscientistsandhistoriansinthefamily.Hisowncareerasanengineerandscientistiscertainlyanunusualone.nHewasamidshipmanintheRoyalSwedishCoast Guard in 1904 was promoted tosublieutenantin1907,Captainin
25、1916,andMajorin1940.HetookcoursesattheRoyalInstitute of Technology where he laterbecameafullprofessor(1924)andgraduatedin1924.HisdoctorateisfromtheUniversityofUppsalain1932.HeworkedinSwedish and German industries as aninventor (ball and roller bearings, electrichammer,)andasaconsultingengineer. Myfr
26、iendsatSAABinTrollhattenSwedengavemesomeofWeibullspapers.SAABisoneofmanycompaniesthatemployedWeibullasaconsultant.BackgroundBackgroundWWaloddialoddi WeibullWeibull (1887-1979) (1887-1979) invented theinvented the WeibullWeibull distribution in1937.distribution in1937. His 1951 paper represents the c
27、ulmination His 1951 paper represents the culmination ( (顶顶峰峰 ) )of his workof his work in reliability analysis.in reliability analysis. The U.S.Air Force recognized the merit of The U.S.Air Force recognized the merit of WeibullsWeibullsmethods and funded his research to 1975. methods and funded his
28、research to 1975. Leonard Johnson at Leonard Johnson at GenralGenral Motors, improved Motors, improved WeibullsWeibulls methods. methods. WeibullWeibull used mean rank values for plotting used mean rank values for plottingbut Johnson suggested the use of median rank values. but Johnson suggested the
29、 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 todeterminethetypeofoceanbottomsediment
30、sandtheirthickness,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),1961.27ofthesep
31、aperswerereportstotheUS 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 t
32、he 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 smallsamplesat
33、Pratt&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
34、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
35、 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 biologicals
36、ystems because ithas an increasing failure rate and candescribewearoutprocesses.Well,thesetwomen,bothintheirlateseventiesatthetime,showedthatthesedistributionsdidnotapplytothem.Theydidnotwearoutbutwerefulloflifeandenergy.Gumbelwentskiingeveryweekend and when I took Dr. and Mrs.WeibulltotheRooseveltH
37、omeinHydeParkonacoldwinterday,herefusedmyofferedarmtohelphimontheicywalkwayssaying:“A little ice and snow never bothered aSwede.”nIn 1941 BOFORS, a Swedish armsfactory, gave him a personal researchprofessorshipinTechnicalPhysicsattheRoyalInstituteofTechnology,Stockholm.nIn1972,theAmericanSocietyofMe
38、chanicalEngineers(4)awardedDr. WeibulltheirgoldmedalcitingProfessorWeibullas“apioneerinthestudyoffracture,fatigue,andreliabilitywhohascontributedtotheliteratureforoverthirty years. His statistical treatment ofstrength and life has found widespreadapplicationinengineeringdesign.”Theawardwas presented
39、 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 storytoldbyfriendsatW
40、rightPattersonAirForce 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 a
41、t 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
42、 hewastheprincipalauthor of both. The latest version of thatHandbook is the fourth edition of The NewWeibullHandbook(6).nProfessorWeibullsproudestmomentcamein1978whenhereceivedtheGreatGoldmedalfromtheRoyalSwedishAcademyofEngineering Sciences, which was personallypresentedtohimbyKingCarlXVIGustavofSw
43、edennHewasdevotedtohisfamilyandwasproudofhisninechildrenandnumerousgrandandgreat-grandchildren.nDr. Weibull was a member of manytechnical societies and worked to thelastdayofhisremarkablelife.HediedonOctober12,1979inAnnecy,France.nTheWeibullDistributionwasfirstpublishedin1939,over60yearsagoandhaspro
44、ventobe 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 Ne
45、w Weibull Handbook touniversitylibrariesinEnglishspeakingcountriesthatrequestthebook.ThecorrespondingSuperSMITHsoftwareisavailable from Wes Fulton in demo versionfreefromhisWebsite.n()BackgroundBackground E.J.Grumbel proved that the Weibull distribution and the smallest extreame value distributions(
46、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 Advantages of WeibullWeibull Analysis Analysis Small SamplesSmall SamplesThe primary advantage of The primary advantage of WeibullWeibull an
47、alysis is the ability to 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 Effect
48、ive component testing. Effective component testing. Graphical AnalysisGraphical AnalysisAnother advantage of Another advantage of WeibullWeibull analysis is that it have a simple analysis is that it have a simple and useful graphical plot. It can be easily generated with and useful graphical plot. I
49、t can be easily generated with cumulative probability paper.cumulative probability paper. Advantages of Advantages of WeibullWeibull Analysis Analysis Application AreasApplication Areas Failure forecasting and prediction,Failure forecasting and prediction, Evaluating corrective action plans,Evaluati
50、ng 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 suppo
51、rt 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
52、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 verif
53、ying 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?”WeibullWeibull Distribution Distribution: : Infant mortality (wear in failures)Infant mortality (wea
54、r in failures): : Independent of age (random failures)Independent of age (random failures): : Wear out failuresWear out failures: : Shape parameterShape parameter: : Scale parameterScale parameterWeibullWeibull Distribution DistributionCumulative Failure PlotCumulative Failure PlotCumulative Failure
55、 PlotCumulative Failure PlotWeibullWeibull Distribution : Distribution : Its statistical propertiesIts statistical propertiesMedianMedianWeibullWeibull Distribution : Distribution : Its statistical propertiesIts statistical propertieswhenwhenthenthenwhenwhenthenthenwhenwhenthenthenwhenwhenthenthenWe
56、ibullWeibull Distribution : Distribution : Its statistical propertiesIts statistical propertiesFrom the above equation, From the above equation, If we set the value of time (t) to t = If we set the value of time (t) to t = then F( then F() = 0.632.) = 0.632.So we can guess that So we can guess that
57、is defined as the age at which is defined as the age at which 63.2% of the units will fail.63.2% of the units will fail.Interpretation of ParametersInterpretation of ParametersAssume that we have 2-types data, failure and Assume that we have 2-types data, failure and suspended. suspended. For scale
58、parameter For scale parameter , , In general the more we have suspended data In general the more we have suspended data the shape parameter , the shape parameter , , hardly change but the scale , hardly change but the scale parameter , parameter , , will be increased. , will be increased. For shape
59、parameter For shape parameter , , : : Implies Infant mortality and we can suspect Implies Infant mortality and we can suspect Interpretation of ParametersInterpretation of Parameters Inadequate burn-in test or screening.Inadequate burn-in test or screening. Production problems, Production problems,
60、misassemblymisassembly, quality control., quality control. Overhaul problemsOverhaul problems Solid state electronic failureSolid state electronic failure: : Implies random failures Implies random failures Maintenance errors, human errorsMaintenance errors, human errors Failures due to natureFailure
61、s due to natureInterpretation of ParametersInterpretation of Parameters Failures due to natureFailures due to nature Mixtures of data from 3 or more failure modesMixtures of data from 3 or more failure modesor different or different s s : : Implies early wear out Implies early wear out Low cycle fatigueLow cycle fatigue Corrosion or erosionCorrosion or erosion: : Implies ageing effects Implies ageing effects Median Rank RegressionMedian Rank Regression Preliminary : Linear RegressionPreliminary : Linear Regression 1.Model : 2.Statistical Assumption :
