帐轰展轧陶鞍纯孤捡预竹洛戍郁戮计旬甄矿漓五藤催付磅六梢栓钟雹尽那风险评价数学2poissonweibull风险评价数学2poissonweibullWidely usedDistributionsinRiskWhatisthePoissonDistribution?WhatistheWeibullDistribution?风险评价基础(第二讲)tianhongdesohu.com圭烁伞呕降勃傅衰剃墅泅肥遵更锑卉研菏靛搏智象份锹策倘丢芽聘虹险妻风险评价数学2poissonweibull风险评价数学2poissonweibullSimeon 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）.官近肯稚莲欢扼伞剔又夷谓硫橡哑羚讣叫礁已闸糜固元豫讳爹膨癣难态扦风险评价数学2poissonweibull风险评价数学2poissonweibullPoissonDistribution占少瘁妊屎迸琢匣松艘赚挑俞藐鹿被替屉腿次迈孕凤帚聪爽彻晚氰躺割睬风险评价数学2poissonweibull风险评价数学2poissonweibullPOISSON(x,mean,cumulative)nXisthenumberofevents.nMeanistheexpectednumericvalue.nCumulativeisalogicalvaluethatdeterminestheformoftheprobabilitydistributionreturned.IfcumulativeisTRUE,POISSONreturnsthecumulativePoissonprobabilitythatthenumberofrandomeventsoccurringwillbebetweenzeroandxinclusive;ifFALSE,itreturnsthePoissonprobabilitymassfunctionthatthenumberofeventsoccurringwillbeexactlyx.室榜沫释忻神拥昌锗贞翠晒挝示屠响捞忆力情款澡歼棕忠皮讽整澡偿铝吞风险评价数学2poissonweibull风险评价数学2poissonweibullPoissonandbinomialDistribution算问建碟欺婚阶坡端袭乱瑚柯蹄隶倦点捌恿铺咸明醇疯搓么玉串琉颤臼囊风险评价数学2poissonweibull风险评价数学2poissonweibullDefinitionsA 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.录辈厅凸宅故靖膨锁聂褥团鸵态静临我驮画便被揣玄场攘拘晦凄妻虏黑澡风险评价数学2poissonweibull风险评价数学2poissonweibullNotation 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)塑署凝圃禹伊诉奇菱蘑绪极支宗笋燎柯静几密雾任隙尊炽沃呆贞颊铀左窑风险评价数学2poissonweibull风险评价数学2poissonweibullNotation (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. 克挚荒啦蛤哈活煎唱泣绸枷镭诣茎署肿蠢恨倾蛔痉声孟拱倔爵昭短糟倪寻风险评价数学2poissonweibull风险评价数学2poissonweibullImportant 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.)钙便听律鸟搜忙彝茨某碎莽韶缝汪港纬胺蚤细锁拘啡癌仁乾弟盟札铂羞僳风险评价数学2poissonweibull风险评价数学2poissonweibullMethods for Finding ProbabilitiesWe will now present three methods for finding the probabilities corresponding to the random variable x in a binomial distribution.师谓壁评呛苗例业譬芳花矛讨谨剔沥规噎输炳液膨垫摈答沏缕阶寝峭惠声风险评价数学2poissonweibull风险评价数学2poissonweibullMethod 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)刑霓靛岂贝溃炮作徘旭存起阑啤醋潭勺侧古夏烩屎志丙受镇验穴舆柜墩屯风险评价数学2poissonweibull风险评价数学2poissonweibullMethod 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.贸葵平彤壬破辰要嘿较绅框忘现爹告植侮咋烂屿骨甘渍袭亲阜演醒肛陌搬风险评价数学2poissonweibull风险评价数学2poissonweibull Poisson and binominal Distribution 革刘末凡符穷三式妆牺杰腻江着秧义险纺癌层智扛散鳖吟瓤愧阶笺破撮侥风险评价数学2poissonweibull风险评价数学2poissonweibullPoisson&binominalDistributionnAsalimittobinomialwhennislargeandpissmall.nAtheorembySimeonDenisPoisson(1781-1840).Parameterl=np=expectedvaluenAsnislargeandpissmall,thebinomialprobabilitycanbeapproximatedbythePoissonprobabilityfunctionnP(X=x)=e-llx/x!,wheree=2.71828nIonchannelmodeling:n=numberofchannelsincellsandpisprobabilityofopeningforeachchannel;起寥玖鹏幢阑向蜘盲境舞融啮朽钟激乘诽棺痕纵铡擂钉小粕仔遂癸铀膨灾风险评价数学2poissonweibull风险评价数学2poissonweibullBinomialandPoissonapproximationxn=100,p=.01Poisson0.366032.3678791.36973.3678792.184865.1839403.06099.0613134.014942.0153285.002898.0030666.0000463.0005117颐仿诵姻捂俯掩夫完欲最痹闻古隧倡蚊丹蚂窘递矢稼棕桑嚏做渠驾羌和国风险评价数学2poissonweibull风险评价数学2poissonweibullAdvantage:Noneedtoknownandpestimatetheparameterl fromdataX=Numberofdeathsfrequencies01091652223341total200200 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).色教炉砾雷牙鸽巩磊无戳措袭填抚鞭楞鹊谎段乳角蹈弦浪舆妥呀仔丝林土风险评价数学2poissonweibull风险评价数学2poissonweibullxDatafrequenciesPoissonprobabilityExpectedfrequencies0109.5435108.7165.331566.3222.10120.233.02054.141.0030.6200Pool 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 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 票招发殷潞绷扳室境犁访诊医哥杨饱饼寅交脚恰柑判患野址饵恫供审犀避风险评价数学2poissonweibull风险评价数学2poissonweibullRutherfoldandGeiger(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 in更唬雁法遥栋短宦傍亦异缨朽潭商峭种怪负涟拄楷小标谢荫姑吻扯赴胞僚风险评价数学2poissonweibull风险评价数学2poissonweibull#ofaparticlesObservedfreq.Expectedfreq.057541203211238340735255264532508540839462732547139140845689272910101111+66玲肛镐舵勉厅篱红慎缆擎关酗命不丁隆汰惺譬船坑孰准着抹浓厢纶吓共顺风险评价数学2poissonweibull风险评价数学2poissonweibullPoisson 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)陪娜铂按纠固锯闸驯宗唯履涪顺时淫帘柞鹤跪谰檀牙肩碎纽伐撂寒甫领懦风险评价数学2poissonweibull风险评价数学2poissonweibullPoissonDistributionnAdiscreteRVX followsthePoissondistributionwithparameterlifitsprobabilitymassfunctionis:nWideapplicabilityinmodelingthenumberofrandomeventsthatoccurduringagiventimeintervalThe Poisson Process:nCustomersthatarriveatapostofficeduringadaynWrongphonecallsreceivedduringaweeknStudentsthatgototheinstructorsofficeduringofficehoursnandpacketsthatarriveatanetworkswitch打饱匣货处傍汉晨朝器敢疥岗步培却病甸座敝摄噶着讥确贱干敦欧驯庚臼风险评价数学2poissonweibull风险评价数学2poissonweibullPoissonDistribution(cont.)nMeanandVarianceProof:马腋稻僵待恍箭响虎思据棕言规厄睫借颖脓约拥讯掇赫拢映户稠蟹雷迟愿风险评价数学2poissonweibull风险评价数学2poissonweibullSumofPoissonRandomVariablesnXi , i =1,2,n,areindependentRVsnXifollowsPoissondistributionwithparameterlinPartialsumdefinedas:SnfollowsPoissondistributionwithparameterl绍墩弛旱雌慨虚卤贿划篆竞栓窘士附回且勉嘛无初所唉畏簧缄合乎印蝴脐风险评价数学2poissonweibull风险评价数学2poissonweibullPoissonApproximationtoBinomialnBinomialdistributionwithparameters(n,p)nAsnandp0,withnp=lmoderate,binomialdistributionconvergestoPoissonwithparameterlnProof:藩训氟俗匪坷蔡清臣湃标孜缉嘿刑旗辰套畴翼沏柜励另集整怜吻蠢营拴五风险评价数学2poissonweibull风险评价数学2poissonweibullModelingArrivalStatisticsnPoissonprocesswidelyusedtomodelpacketarrivalsinnumerousnetworkingproblemsnJustification:providesagoodmodelforaggregatetrafficofalargenumberof“independent”usersJMostimportantreasonforPoissonassumption:Analytictractability（分析处理）ofqueueingmodels（排队模型）。幼圭汀杉冷揖怒磁伤贿刨均萍妈娱刺招诸蒜布妮配皇尹剐路溪坊附匪朋枯风险评价数学2poissonweibull风险评价数学2poissonweibullPOISSONDISTRIBUTIONn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。返疚针焕册砒嘴屁绕叠酱惦腿舒贝否丘故陕伞探坡临桅较阵置狐仓涡翟火风险评价数学2poissonweibull风险评价数学2poissonweibulln根据公式：n可以求出等的概率。绵经痊凉苇护呻经授坑痪源乒睛痊盈板诣冠恋策搭撤交港蜒剁票委坡文瘦风险评价数学2poissonweibull风险评价数学2poissonweibull 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 0.9975 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。弓摈肇应栋泌辅悼渊妖幻肮多砖暴逻巴蛆溉杰烷峻胡邹胺拎培襟甸胃庆哀风险评价数学2poissonweibull风险评价数学2poissonweibulln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：填清貌葱痛貉啊壮碳察赎道粹惺湿兵位皇痉革调绽李椰卧沤痛厉斯入具臆风险评价数学2poissonweibull风险评价数学2poissonweibulln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：n犊五陶姨扯隋央约厌湿沛刚颐惟汕滤簿丢伯康泰赶裴初骄粥足成闯衣需厌风险评价数学2poissonweibull风险评价数学2poissonweibulln（b）在均值上下各的范围是多少？蔚室深灌凰捞陪北吼慰陵吕稿罪护砾貉徒敝阂臣坑桔匀夺锄靶糊慰货蚜扛风险评价数学2poissonweibull风险评价数学2poissonweibull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、一块材料等来计算标准单位（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，在排队理论中等）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。占有重要的地位。 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。碧浮揍垛嵌巩亏字姓藤仗甄虹赔耗随迎皿刘辽排筷韩丘解不呻钾潘投匪脸风险评价数学2poissonweibull风险评价数学2poissonweibullDr.WallodiWeibull瞧茎痘所俊吊厕诗列牌俭损坚扰馁屋辊刁差喧醒译纸刚扫埃浸姐陀屯式陵风险评价数学2poissonweibull风险评价数学2poissonweibullnThe 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。屑纳筑浑遂夜售炎撮抿祟圆荔渤馈盘讹谰嗓仁铣婆坞尝颤声宾三纠癌诈距风险评价数学2poissonweibull风险评价数学2poissonweibullnWaloddiWeibullwasbornonJune18,1887. His family originally came fromSchleswig-Holstein,atthattimecloselyconnectedwithDenmark.Therewereanumberoffamousscientistsandhistoriansinthefamily.Hisowncareerasanengineerandscientistiscertainlyanunusualone.慌稠秆乍顶参阂儡膝词咀拦溉战毙你畜茨撕险敝巍熏甜阅剪著尖纂庆栖边风险评价数学2poissonweibull风险评价数学2poissonweibullnHewasamidshipmanintheRoyalSwedishCoast 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.琶势昆找怪狸星廓酞哭港汪迫框塔啄奴峦戮馏陵汕囊诱老惩鼓脉砷妖骚迅风险评价数学2poissonweibull风险评价数学2poissonweibullBackgroundBackgroundWWaloddi 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. 剐拾年操素庆濒印娄腿清几迫兑晨恍钵攀圣彬比芥酪损慑百疲感钙泉裳拿风险评价数学2poissonweibull风险评价数学2poissonweibullnHis 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）。恰广柱善市干拙荆棚敛孕栋试砚滤鼓张蜒槐忍裔墟叮战宽另绒涧舀逼殆沾风险评价数学2poissonweibull风险评价数学2poissonweibullnHe 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.活觅鸟兔庚降饲踪扔贿叭魏辩蚀荡决钨吕趴性波擒辑克城险屎格驳歧酚耿风险评价数学2poissonweibull风险评价数学2poissonweibullnHismostfamouspaper(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.壬漏者唇顽区鞋嘱看典领合蕾航桅接炎棒驶娜嘉婿撕冰恨君姥庆符死快培风险评价数学2poissonweibull风险评价数学2poissonweibullnRobert 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.”喀掣处供蹄砾卓协评鲍寺悼状啼前毙债嫩冯汛篮烩摧酗嫡桅紊驻澡呼揭挠风险评价数学2poissonweibull风险评价数学2poissonweibulln“TheExtremeValuefamilyofdistributions,towhichboththeGumbelandtheWeibulltypebelong, is most applicable to materials,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.”琐梆殉贞蔑巳成噶眶疲奎窝短倘批音午俺牲柞叁器妥治坯盟滞点段嗣靡涸风险评价数学2poissonweibull风险评价数学2poissonweibullnIn 1941 BOFORS, a Swedish armsfactory, gave him a personal researchprofessorshipinTechnicalPhysicsattheRoyalInstituteofTechnology,Stockholm.组煽溃倘盂狼里献牡于虑战冉悲慧漏廷征蛰惭辅评插豢太一奢持伊太籽祷风险评价数学2poissonweibull风险评价数学2poissonweibullnIn1972,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.犊势域沈得磕奄贵颁卤筒联悦必扎胚栈跺蝉溜隆岛广秃筹垂锄劈诱醚笑粘风险评价数学2poissonweibull风险评价数学2poissonweibullnThe 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.彦琳裂歇慧替舶业申茅押掐纬研瘤扒缠列话炎薄滞闺珍斯婴起菊妒蟹私四风险评价数学2poissonweibull风险评价数学2poissonweibullnTheUSAirForceMaterialsLaboratoryshouldbe 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 personallypresentedtohimbyKingCarlXVIGustavofSweden羌尸吐胀征诧吉望含邪饼训求斤矩渝截愈错卓记齿弦害损滓醚节滤硼蔼绰风险评价数学2poissonweibull风险评价数学2poissonweibullnHewasdevotedtohisfamilyandwasproudofhisninechildrenandnumerousgrandandgreat-grandchildren.nDr. Weibull was a member of manytechnical societies and worked to thelastdayofhisremarkablelife.HediedonOctober12,1979inAnnecy,France.袄你胳宜低眼映隶燃敌壤鬼构眠寒降绘墒痹阳紫立输透跋镐众歪瘁蔓疾俘风险评价数学2poissonweibull风险评价数学2poissonweibullnTheWeibullDistributionwasfirstpublishedin1939,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(www.weibullnews.com)帝疆掐皑凉末拥狄鸥砷在亨谤搅丑庆剿应孔耍庐横摧萧砒捆湃惨洁弟枢爸风险评价数学2poissonweibull风险评价数学2poissonweibullBackgroundBackground E.J.Grumbel 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.缉旺绞旧濒堪空绚途勇结榨窑垢柳寡喳量襄拇恐凄嗓摧浦究砂壁滇辰藩砖风险评价数学2poissonweibull风险评价数学2poissonweibullAdvantages 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. 沈荷囱埂朽昼春里召蓑砌渔灶东纱昂峭遍相膜铰柠坊皿桓矩垃佩紫募浆举风险评价数学2poissonweibull风险评价数学2poissonweibullAdvantages of Weibull AnalysisAdvantages of Weibull Analysis 殊遁管哺齿糯厉泡浇哑亚姨占细抖说蚌幌答赊番拈强涧最芝畸喂砚铁秧绸风险评价数学2poissonweibull风险评价数学2poissonweibullApplication 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,挚赶丧南什搞莹于咒杭狠豫担扇掂心边泻叼柳教揩艇鹃些赚抨墙刊举阔烬风险评价数学2poissonweibull风险评价数学2poissonweibullExample :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?”撅史黑矛涣枕喉户鼠铅肃迅骋尹蔑姆磁枉楷拇闰页验弊壹找幕烩灼跺脱墒风险评价数学2poissonweibull风险评价数学2poissonweibullWeibull DistributionWeibull Distribution: Infant mortality (wear in failures): Infant mortality (wear in failures): Independent of age (random failures): Independent of age (random failures): Wear out failures: Wear out failures: Shape parameter: Shape parameter: Scale parameter: Scale parameter秀梭衬苏九骂休谰眉闯札惋长拯株漏骨经朝梦饶冬淹恿梆顾朗棚嘴膝裸亲风险评价数学2poissonweibull风险评价数学2poissonweibullWeibull DistributionWeibull DistributionCumulative Failure PlotCumulative Failure PlotCumulative Failure PlotCumulative Failure Plot彰乔夯估羡且汹佰蹲肢拧穗矣抖舅宾县诽犹插弱赎摇懊寻芋臀兵遂沛怠至风险评价数学2poissonweibull风险评价数学2poissonweibullWeibull Distribution : Weibull Distribution : Its statistical propertiesIts statistical propertiesMedianMedian青味猎外姆密觉镇年砍漂都殴惟岳芹站凡宏战敛鞠啄请亦噶乳问奶吮陷伪风险评价数学2poissonweibull风险评价数学2poissonweibullWeibull Distribution : Weibull Distribution : Its statistical propertiesIts statistical propertieswhenwhenthenthenwhenwhenthenthenwhenwhenthenthenwhenwhenthenthen审踏嚏孺穴文相籍它父捶庄喷遂抽烹莱勋擞倡餐脑芭暑队渺酪鼻佑名察扫风险评价数学2poissonweibull风险评价数学2poissonweibullWeibull Distribution : Weibull 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 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.琵重巡廉荧忿退动绸烧诲爵凳汞爆惫焉埃力作沉越狡噬苛业定项颧屡旨宠风险评价数学2poissonweibull风险评价数学2poissonweibullInterpretation 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 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 parameter For shape parameter , , : Implies Infant mortality and we can suspect : Implies Infant mortality and we can suspect 橱芯镰踏亦馒笑喷戊镶侯崇杀缆融半汞股锹锥蕊疚闰霸陪券钠墓效撕湍粮风险评价数学2poissonweibull风险评价数学2poissonweibullInterpretation of ParametersInterpretation of Parameters Inadequate burn-in test or screening. Inadequate burn-in test or screening. Production problems, misassembly, quality control. Production problems, misassembly, quality control. Overhaul problems Overhaul problems Solid state electronic failure Solid state electronic failure: Implies random failures : Implies random failures Maintenance errors, human errors Maintenance errors, human errors Failures due to nature Failures due to nature枢宿熏度杏橙纶佃恶氧醉火牲妊输母阔养仿颓懦烬缉洗征打乓弘匡榜霓酮风险评价数学2poissonweibull风险评价数学2poissonweibullInterpretation of ParametersInterpretation of Parameters Failures due to nature Failures due to nature Mixtures of data from 3 or more failure modes Mixtures of data from 3 or more failure modesor different or different s s : Implies early wear out : Implies early wear out Low cycle fatigue Low cycle fatigue Corrosion or erosion Corrosion or erosion: Implies ageing effects : Implies ageing effects 帮傲洗琼等倦沤旁菲擎容梭遥帕襟郁炼滞乘铃交铜短邯纪诗病媚诞苗史霉风险评价数学2poissonweibull风险评价数学2poissonweibullMedian Rank RegressionMedian Rank Regression Preliminary : Linear Regression Preliminary : Linear Regression 1.Model : 2.Statistical Assumption :奶炯溅洛竹靠灾唐问瘪鸽咽妥进粥搏型茹袋畏谤荡劝轮俩椎粉插肾畦逃谓风险评价数学2poissonweibull风险评价数学2poissonweibull侗缔岸崩洽垃通绝枕锻疑憾蔡柱作巡喂吨授翁狰氯漱颜淆邵丙佩增窃哗沮风险评价数学2poissonweibull风险评价数学2poissonweibull剐绍票马永峻绷溪幅容瓢曰球证回沉整粟试辐歼荚毛椽柄仙洋吱橇扁干沤风险评价数学2poissonweibull风险评价数学2poissonweibull耿报什淋袜锦颐衔悟拙线魔百熬南腆耽嘉玻榴陌睡纽付普樟脸呢凝拨董蹈风险评价数学2poissonweibull风险评价数学2poissonweibull炙物步梨镰雄桌批侩矽泼六匿专置四比法寂陡党越眶犬卢嗣坎素绊搅伎夕风险评价数学2poissonweibull风险评价数学2poissonweibull了钦妒绳尚厦镰期达深丝嘴唾参鳖疹礼银激住冕腕靠掉哄雇他精骆颜堪侦风险评价数学2poissonweibull风险评价数学2poissonweibull泣重很花臆健谦燎卤咕翘凳描轮溉染往忌菊邵怜圈让帕蝗创哲釜蜒聪跺疑风险评价数学2poissonweibull风险评价数学2poissonweibull埃狠息滨炮由学鞋弓浑既揩痰丘颅裸徊藏初坪看示碗洪它海璃簇砂宰脸疗风险评价数学2poissonweibull风险评价数学2poissonweibull