风险评价数学poissonweibull.ppt
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1、Widely used Distributions in Risk,What is the Poisson Distribution? What is the Weibull Distribution?,风险评价基础 ( 第二讲) ,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
2、(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 determi
3、nes 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 occur
4、ring 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
5、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;
6、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
7、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
8、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 probabilitie
9、s 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 Ta
10、ble 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 bino
11、mial 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
12、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(
13、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
14、 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 (Magneti
15、c 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
16、 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
17、 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 a
18、s: 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 pack
19、et 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
20、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:,检查前提假设是否成
21、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。 确定变量,求出值; 求对应个别K的泊松分布概率; 求若干个K的泊松分布概率的总和; 求泊松分布的均值和方差; 画出概率分布和累积分布图。,Dr. Wallodi Weibull,The Weibull distribution is by far the worlds most popular statistical model for life data(寿命数据). It is also used in man
22、y 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 wa
23、s 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 S
24、wedish 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 Swedis
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