風(fēng)險評價數(shù)學(xué)poissonweibull.ppt

Widely used Distributions in Risk,What is the Poisson Distribution? What is the Weibull Distribution?,風(fēng)險評價基礎(chǔ) ( 第二講) ,Simeon Denis Poisson,“Researches on the probability of criminal and civil verdicts” 1837（犯罪和民法裁決） . looked at the form of the binomial distribution when the number of trials was large（試驗的次數(shù)較大時）. He derived the cumulative Poisson distribution as the limiting case of the binomial when the chance of success tend to zero（成功的機會趨于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 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 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 & binominal Distribution,As a limit to binomial when n is large and p is small. A theorem by Simeon Denis Poisson(1781-1840). Parameter l= np= expected value As n is large and p is small, the binomial probability can be approximated by the Poisson probability function P(X=x)= e-l lx / x! , where e =2.71828 Ion channel modeling : n=number of channels in cells and p is probability of opening for each channel;,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(騎兵部隊).,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,Rutherfold and Geiger (1910) 盧瑟福和蓋革,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 ) (核磁共振檢查),Other related application in,Poisson process for modeling number of event occurrences in a spatial( 空間的) or temporal domain(時間的區(qū)域),Homogeneity(同一性) : rate of occurrence is uniform Independent occurrence in non-overlapping areas(非疊加),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 instructors 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 p0, 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（分析處理） of queueing models（排隊模型）。,POISSON DISTRIBUTION,例題：如果電話號碼本中每頁的錯誤個數(shù)為2.3個，K為每頁中錯誤數(shù)目的隨機變量。（a）畫出它的概率密度和累積分布圖；（b）求足以滿概括50%頁數(shù)中差錯誤的K 。,根據(jù)公式： 可以求出等的概率。,關(guān)于概率分布曲線以及累計概率分布曲線的繪制和分析的問題: (1)離散分布； (2)其代表的具體意義。,例題：某單位每月發(fā)生事故的情況如下： 每月的事故數(shù) 0 1 2 3 4 5 頻 數(shù)（月數(shù)） 27 12 8 2 1 0 注意：一共是50個月的統(tǒng)計資料：,根據(jù)如上的數(shù)據(jù)，認(rèn)為 （a）最有可能的是每月發(fā)生一次事故，這正確嗎？ （b）在均值上下各的范圍是多少？ （a）解：每月發(fā)生一次事故概率為：,（b）在均值上下各的范圍是多少？,應(yīng)用泊松分布解題的步驟如下：,檢查前提假設(shè)是否成立。最主要的條件是在每一標(biāo)準(zhǔn)單位內(nèi)所指的事件發(fā)生的概率是常數(shù)；泊松分布用來計算標(biāo)準(zhǔn)單位（一張照片、一只機翼、一塊材料等等）內(nèi)的缺陷數(shù)、交通死亡人數(shù)等等，在排隊理論中占有重要的地位。 確定變量，求出值； 求對應(yīng)個別K的泊松分布概率； 求若干個K的泊松分布概率的總和； 求泊松分布的均值和方差； 畫出概率分布和累積分布圖。,Dr. Wallodi Weibull,The Weibull distribution is by far the worlds most popular statistical model for life data（壽命數(shù)據(jù)）. It is also used in many other applications, such as weather forecasting and fitting data of all kinds（數(shù)據(jù)擬合）. 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 Weibulls 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 (頂峰 ) of his work in reliability analysis. The U.S.Air Force recognized the merit of Weibulls methods and funded his research to 1975. Leonard Johnson at Genral Motors, improved Weibulls 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（地震波技術(shù)來測量沉積巖的種類和厚度）。,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 Weibulls 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 Universitys 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 Weibulls 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. (),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