Widely used Distributions in Risk,What is the Poisson Distribution? What is the Weibull Distribution?,风险评价基础 ( 第二讲)

[email protected],Simeon Denis Poisson,“Researches on the probability of criminal and civil verdicts” 1837（犯罪和民法裁决） . looked at the form of the binomial distribution when the number of trials was large（试验的次数较大时）. He 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）.,Poisson Distribution,POISSON(x,mean,cumulative),X is the number of events. Mean is the expected numeric value. Cumulative is a logical value that determines the form of the probability distribution returned. If cumulative is TRUE, POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive; if FALSE, it returns the Poisson probability mass function that the number of events occurring will be exactly x.,Poisson and binomial Distribution,Definitions,A 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 doesn’t 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 Distributions,S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so,P(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 Hints,Be sure that x and p both refer to the same category being called a success.,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 Probabilities,We 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,where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p),Method 2: Using Table A-1 in Appendix A,Part 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 ,Binomial and Poisson approximation,Advantage: No need to know n and p estimate the parameter l from data,,200 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. Pearson’s 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,Rutherfold and Geiger (1910) 卢瑟福和盖革,Polonium(钚) source placed a short distance from a small screen. For each of 2608 eighth-minute intervals, they recorded the number of alpha particles impinging on the screen,,Medical Imaging : X-ray, PET scan (positron emission tomography), MRI (Magnetic Resonance Imaging ) (核磁共振检查),Other related application in,,Poisson 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 uniform Independent occurrence in non-overlapping areas(非叠加),,,Poisson Distribution,A discrete RV X follows the Poisson distribution with parameter l if its probability mass function is: Wide applicability in modeling the number of random events that occur during a given time interval – The Poisson Process: Customers that arrive at a post office during a day Wrong phone calls received during a week Students that go to the instructor’s office during office hours … and packets that arrive at a network switch,Poisson Distribution (cont.),Mean and Variance Proof:,Sum of Poisson Random Variables,Xi , i =1,2,…,n, are independent RVs Xi follows Poisson distribution with parameter li Partial sum defined as: Sn follows Poisson distribution with parameter l,Poisson Approximation to Binomial,Binomial distribution with parameters (n, p) As n→∞ and p→0, with np=l moderate, binomial distribution converges to Poisson with parameter l,Proof:,Modeling Arrival Statistics,Poisson process widely used to model packet arrivals in numerous networking problems Justification: provides a good model for aggregate traffic of a large number of “independent” users Most important reason for Poisson assumption: Analytic tractability（分析处理） of queueing models（排队模型）。,POISSON DISTRIBUTION,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 。,根据公式： 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。,,例题：某单位每月发生事故的情况如下： 每月的事故数 0 1 2 3 4 5 频 数（月数） 27 12 8 2 1 0 注意：一共是50个月的统计资料：,根据如上的数据，认为 （a）最有可能的是每月发生一次事故，这正确吗？ （b）在均值上下各的范围是多少？ （a）解：每月发生一次事故概率为：,,,,,（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：,①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，在排队理论中占有重要的地位。 ②确定变量，求出λ值； ③求对应个别K的泊松分布概率； ④求若干个K的泊松分布概率的总和； ⑤求泊松分布的均值和方差； ⑥画出概率分布和累积分布图。,,Dr. Wallodi Weibull,The Weibull distribution is by far the world’s most popular statistical model for life data（寿命数据）. It is also used in many other applications, such as weather forecasting and fitting data of all kinds（数据拟合）. Among all statistical techniques it may be employed for engineering analysis with smaller sample sizes than any other method. Having researched and applied this method for almost half a century。,Waloddi Weibull was born on June 18, 1887. His family originally came from Schleswig-Holstein, at that time closely connected with Denmark. There were a number of famous scientists and historians in the family. His own career as an engineer and scientist is certainly an unusual one.,He was a midshipman in the Royal Swedish Coast Guard in 1904 was promoted to sublieutenant in 1907, Captain in 1916, and Major in 1940. He took courses at the Royal Institute of Technology where he later became a full professor (1924) and graduated in 1924. His doctorate is from the University of Uppsala in 1932. He worked in Swedish and German industries as an inventor (ball and roller bearings, electric hammer,) and as a consulting engineer. My friends at SAAB in Trollhatten Sweden gave me some of Weibull’s papers. SAAB is one of many companies that employed Weibull as a consultant.,?Background,?Waloddi Weibull (1887-1979) invented the Weibull distribution in1937. ? His 1951 paper represents the culmination (顶峰 ) of his work in reliability analysis. ? The U.S.Air Force recognized the merit of Weibull’s methods and funded his research to 1975. ? Leonard Johnson at Genral Motors, improved Weibull’s methods. Weibull used mean rank values for plotting but Johnson suggested the use of median rank values.,His first paper was on the propagation of explosive wave in 1914. He took part in expeditions to the Mediterranean, the Caribbean, and the Pacific ocean on the research ship “Albatross” where he developed the technique of using explosive charges to determine the type of ocean bottom sediments and their thickness, just as we do today in offshore oil exploration（地震波技术来测量沉积岩的种类和厚度）。,He published many papers on strength of materials, fatigue, rupture in solids, bearings, and of course, the Weibull distribution. The author has identified 65 papers to date plus his excellent book on fatigue analysis (1), 1961. 27 of these papers were reports to the US Air Force at Wright Field on Weibull analysis. (Most of these reports to WPAFB are no longer available even from NTIS. The author would appreciate copies of Weibull’s papers from the WPAFB files.) Dr. Weibull was a frequent visitor to WPAFB.,His most famous paper (2) presented in the USA, was given before the ASME in 1951, using seven case studies with Weibull distributions. Many, including the author, were skeptical that this method of allowing the data to select the most appropriate distribution from the broad family of Weibull distributions would work. However the early success of the method with very small samples at Pratt & Whitney Aircraft could not be ignored. Further, Dorian Shainin, a consultant for Pratt & Whitney, strongly encouraged the use of Weibull analysis. The author soon became a believer.,Robert Heller (3) spoke at the 1984 Symposium to the Memory of Waloddi Weibull in Stockholm, Sweden and said, “In 1963, at the invitation of the Professor Freudenthal, he became a Visiting Professor at Columbia University’s Institute for the Study of Fatigue and Reliability. I was with the Institute at that time and got to know Dr. Weibull personally. I learned a great deal from him and from Emil Gumbel and from Freudenthal, the three founders of Probabilistic Mechanics of Structures and Materials. It was interesting to watch the friendly rivalry between Gumbel, the theoretician and the two engineers, Weibull and Freudenthal.”,“The Extreme Value family of distributions, to which both the Gumbel and the Weibull type belong, is most applicable to materials, structures and biological systems because it has an increasing failure rate and can describe wear out processes. Well, these two men, both in their late seventies at the time, showed that these distributions did not apply to them. They did not wear out but were full of life and energy. Gumbel went skiing every weekend and when I took Dr. and Mrs. Weibull to the Roosevelt Home in Hyde Park on a cold winter day, he refused my offered arm to help him on the icy walkways saying: “ A little ice and snow never bothered a Swede.”,In 1941 BOFORS, a Swedish arms factory, gave him a personal research professorship in Technical Physics at the Royal Institute of Technology, Stockholm.,In 1972, the American Society of Mechanical Engineers (4) awarded Dr. Weibull their gold medal citing Professor Weibull as “a pioneer in the study of fracture, fatigue, and reliability who has contributed to the literature for over thirty years. His statistical treatment of strength and life has found widespread application in engineering design.” The award was presented by Dr. Richard Folsom, President of ASME, and President of Rensselaer Polytechnic Institute when the author was a student there. By coincidence the author received the 1988 ASME gold medal for statistical contributions including advancements in Weibull analysis.,The author has an unconfirmed story told by friends at Wright Patterson Air Force Base that Dr. Weibull was in a great state of happiness on his last visit to lecture at the Air Force Institute of Technology in 1975 as he had just been married to a pretty young Swedish girl. He was 88 years old at the time. His first wife has passed on earlier. It was on this trip that the photo above was taken at the University of Washington where he also lectured.,The US Air Force Materials Laboratory should be commended for encouraging Waloddi Weibull for many years with research contracts. The author is also indebted to WPAFB for contracting the original USAF Weibull Analysis Handbook (5) and Weibull video training tape, as he was the principal author of both. The latest version of that Handbook is the fourth edition of The New Weibull Handbook (6). Professor Weibull’s proudest moment came in 1978 when he received the Great Gold medal from the Royal Swedish Academy of Engineering Sciences, which was personally presented to him by King Carl XVI Gustav of Sweden,He was devoted to his family and was proud of his nine children and numerous grand and great-grandchildren. Dr. Weibull was a member of many technical societies and worked to the last day of his remarkable life. He died on October 12, 1979 in Annecy, France.,The Weibull Distribution was first published in 1939, over 60 years ago and has proven to be invaluable for life data analysis in aerospace, automotive, electric power, nuclear power, medical, dental, electronics, every industry. Yet the author is frustrated that only three universities in the USA teach Weibull analysis. To encourage the use of Weibull analysis the author provides free copies of The New Weibull Handbook to university libraries in English speaking countries that request the book. The corresponding SuperSMITH software is available from Wes Fulton in demo version free from his Website. (www.weibullnews.com),?Background,? 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.,?Advantages of Weibull Analysis,?Small Samples The primary advantage of Weibull analysis is the ability to provide failure analysis and failure forecasts accurately with small samples. Furthermore, small samples also allow cost Effective component testing. ? Graphical Analysis Another advantage of Weibull analysis is that it have a simple and useful graphical plot. It can be easily generated with cumulative probability paper.,?Advantages of Weibull Analysis,?Application Areas,? Failure forecasting and prediction,,? Evaluating corrective action plans,,? Engineering change substantiation,,? Maintenance planning and cost effective replacement strategies,,? Spare parts forecasting,,? Warranty analysis and support cost predictions,,?Example :,? In a certain project, “How many failures will we have in the next six month or a year?” ? To do a scheduled maintenance or prepare spares, “How many units will be needed for doing overhauling in the near future?” ? After an engineering change, “How many units must be tested for verifying that the old failure mode is eliminated or improved with confidence level?”,?Weibull Distribution,: Infant mortality (wear in failures),: Independent of age (random failures),: Wear out failures,: Shape parameter,: Scale parameter,?Weibull Distribution,,,,,Cumulative Failure Plot,?Weibull Distribution : It’s statistical properties,Median,?Weibull Distribution : It’s statistical properties,when,then,when,then,when,then,when,then,?Weibull Distribution : It’s statistical properties,From the above equation, If we set the value of time (t) to t = ? then F(?) = 0.632. So we can guess that ? is defined as the age at which 63.2% of the units will fail.,?Interpretation of Parameters,Assume that we have 2-types data, failure and suspended. ? For scale parameter ?, In general the more we have suspended data the shape parameter , ? , hardly change but the scale parameter , ? , will be increased. ? For shape parameter ?,,: Implies Infant mortality and we can suspect,?Interpretation of Parameters,? Inadequate burn-in test or screening.,Production problems, misassembly, quality control.,? Overhaul problems,? Solid state electronic failure,: Implies random failures,? Maintenance errors, human errors,? Failures due to nature,?Interpretation of Parameters,? Failures due to nature,Mixtures of data from 3 or more failure modes or different ?s,: Implies early wear out,? Low cycle fatigue,? Corrosion or erosion,: Implies ageing effects,?Median Rank Regression,? Preliminary : Linear Regression,1.Model : 2.Statistical Assumption :,