工程方法概率分布介绍课件.ppt

1、Page 1Page 2What is a Probability Distribution?什么是概率分布什么是概率分布?Experiment,Sample Space,Event 实验,样本空间,事件Random Variable,Probability Functions(pmf,pdf,cdf)随机变量,概率函数Discrete Distributions离散分布离散分布Binomial Distribution 二项式分布Poisson Distribution 泊松分布.Hypergeometric distribution 超几何分布Continuous Distribution

2、s连续分布连续分布Normal Distribution 正态分布Uniform distribution 均匀分布Exponential distribution 指数分布Logarithmic normal distribution 对数正态分布Weibull distribution 威布尔分布Sampling Distributions样本分布样本分布Z Distribution Z 分布t Distribution t 分布c2 Distribution c2 分布F Distribution F 分布Page 3As we progress from description of

3、data towards inference of data,an important concept is the idea of a probability distribution.当我们从描述性数据进步到推论性数据时,一个重要的内容就是概率分布的概念.To appreciate the notion of a probability distribution,we need to review various fundamental concepts related to it:为了解概率分布的概念,我们需要复习各种基本相关概念:Experiment,Sample Space,Even

4、t实验,样本空间,事件Random Variable 随机变量.What do we mean by inference of data?Page 4Experiment实验实验An experiment is any activity that generates a set of data,which may be numerical or not numerical.实验是产生一系列数据的行为,数据有可能是数字的或非数字的.1,2,.,6(a)Throwing a dice掷骰子Experiment generates numerical/discrete dataPinsStainsR

5、ejectAccept(b)Inspecting for stain marks检检查污点印记查污点印记Experiment generates attribute dataPins(c)Measuring shaft 测量 轴径10.53 mm10.49 mm10.22 mm10.29 mm11.20 mmExperiment generates continuous data实验产生数字实验产生数字/离散数据离散数据实验产生计数性数据实验产生计数性数据实验产生连续性数据实验产生连续性数据Page 5Random Experiment 随机实验随机实验If we throw the dice

6、 again and again,or produce many shafts from the same process,the outcomes will generally be different,and cannot be predicted in advance with total certainty.如果我们掷子一次由一次,或从相同工序生产许多轴,结果会是不同的.不能完全提前预测.An experiment which can result in different outcomes,even though it is repeated in the same manner e

7、very time,is called a random experiment.一个实验导致不同的结果,即使它是每次以相同方式,这叫做随机实验Page 6Sample Space样本空间样本空间The collection of all possible outcomes of an experiment is called its sample space.收集实验的所有可能结果称为样本空间Event事件事件An outcome,or a set of outcomes,from a random experiment is called an event,i.e.it is a subse

8、t of the sample space.一个结果,或一套结果,从一个随机实验出来的称为事件,也就是样本空间的子集Page 7Event事件事件Example例 1:Some events from tossing of a dice.从掷骰子的一些事件.Event 事件1:the outcome is an odd number 结果是奇数Event事件 2:the outcome is a number 4 大于4的结果Example例 2:Some events from measuring shaft:从测量轴径的一些事件Event事件 1:the outcome is a diam

9、eter mean直径大于平均值Event 事件2:the outcome is a part failing specs.未通过规格的结果.E2=x USL E2=5,6 E1=1,3,5 E1=x mPage 8Random Variable随机变量随机变量From a same experiment,different events can be derived depending on which aspects of the experiment we consider important.从一个相同的实验,由于我们认为重要的实验方面不同而产生不同的结果In many cases,i

10、t is useful and convenient to define the aspect of the experiment we are interested in by denoting the event of interest with a symbol(usually an uppercase letter),e.g.:许多方面,它是很有用和方便的定义我们感兴趣的实验方面,通过一个大写的字母表示.举例说明:Let X be the event“the number of a dice is odd”.用X代表事件”骰子的数字是奇数”Let W be the event“the

11、shaft is within specs.”.用W代表事件”轴径尺寸在规格内”Page 9Random Variable随机变量随机变量We have defined a function that assigns a real number to an experimental outcome within the sample space of the random experiment.我们定义了一个函数,其代表了一个在随机实验的样本空间的一个真实实验数字This function(X or W in our examples)is called a random variable b

12、ecause:函数(例子中的X 或W)称为随机变量,是因为:The outcomes of the same event are clearly uncertain and are variable from one outcome to another一个事件的发生结果是明显不定的,是同另一个结果相异的.Each outcome has an equal chance of being selected.每一个结果有相同被选择的机会.PinsMeasuring shaft X=Parts out of specs.(LSL=8 mm,USL=10 mm)0.,7.99998,7.99999,

13、8,8,00001,9.99999,10,10.00001,10.00002,LSLUSLPage 10Probability概率概率To quantify how likely a particular outcome of a random variable can occur,we typically assign a numerical value between 0 and 1(or 0 to 100%).为量化一个随机变量的指定结果发生的可能性,我们指定一个数字介于0和1之间(或0100%)This numerical value is called the probability

14、 of the outcome.这个数字称为结果的概率There are a few ways of interpreting probability.A common way is to interpret probability as a fraction(or proportion)of times the outcome occurs in many repetitions of the same random experiment.有几种方式解释概率.一般的方式是解释概率为在许多相同实验重复后发生的分数(或比例)次数This method is the relative freque

15、ncy approach or frequentist approach to interpreting probability.这种方法概率解释的相对频率模拟或单位频率模拟Page 11Probability Distribution概率分布概率分布When we are able to assign a probability to each possible outcome of a random variable X,the full description of all the probabilities associated with the possible outcomes i

16、s called a probability distribution of X.当我们能够表明一个随机变量的某一个可能结果的概率,则整个可能结果的概率的描述称为X的概率分布A probability distribution is typically presented as a curve or plot that has:一个概率分布被代表为一个曲线或点应有:All the possible outcomes of X on the horizontal axisX的所有的可能结果在水平轴线上The probability of each outcome on the vertical

17、axis每一个结果的概率在纵轴上Page 12随机现象 随机试验 样本点、样本空间 语言表示 事件的表示 集合表示 事件的特征 包含、相等 随机事件 事件间的关系 互斥 事件的运算：对立、并、交、差 Page 13Normal DistributionExponential DistributionUniform DistributionBinomial DistributionDiscrete Probability Distributions(Theoretical)离散概率分布离散概率分布(理论上理论上)Continuous Probability Distributions(Theor

18、etical)连续概率分布连续概率分布(理论上理论上)Page 14Created from actual observations.Usually represented as histograms.根据实际观测得来,通常用直方图代表Empirical distributions,like theoretical distributions,apply to both discrete and continuous distributions.经验分布,象理论上的分布,适用于离散和连续分布.Page 15Three common important characteristics:三个常用重

19、要Shape-defines nature of distribution形状 -定义分布的自然性Center-defines central tendency of data中心 -定义中心趋势的数据Spread分布(或离散,或刻度)-defines dispersion of data(or Dispersion,or Scale)定义数据的离散Exponential DistributionUniform Distribution统一分布统一分布指数分布指数分布Page 16 xexfx22121mShape形状形状lDescribes how the probabilities of

20、all the possible outcomes are distributed.l描述所有可能结果可能性的分布lCan be described mathematically with an equation called a probability function,e.g:l可以用一个概率函数数字表示,举例说明Probability function概率函数Lowercase letter represents a specific value of random variable X小字母代表随机变量X某一个特定值 f(x)means P(X=x)Page 1700f(t)1a2a3

21、ab=4210.5Probability Functions概率函数概率函数For a discrete distribution,对于一个离散分布f(x)called is the probability f(x)称为概率集中:mass function(pmf),e.g.:函数,举例说明For a continuous distribution,对于一个连续分布f(x)is called the probability f(x)称为概率密度density function(pdf),e.g.:函数举例说明 n,0,1,2,xp1pxnxPxnx 0,1tettftbabbabPage 18

22、Binomial DistributionNormal DistributionThe total probability for any distribution sums to 1.任何分布的全部概率总和为1In a discrete distribution,probability is representedas height of the bar.在一个离散分布,概率用柱状表示In a continuous distribution,probability is representedas area under the curve(pdf),between two points.在一

23、个连续分布,概率用曲线下两点间面积表示Page 19Probability of An Exact Value Under PDF is Zero!PDF下一个准确值的概率是零下一个准确值的概率是零For a continuous random variable,the probability of an exact value occurring is theoretically 0 because a line on a pdf has 0 width,implying:对于一个连续随机变量,一个准确值发生的概率理论上是0,是因为PDF上一条线的宽度是0”.意味着:In practice,

24、if we obtain a particular value,e.g.12.57,of a random variable X,how do we interpret the probability of 12.57 happening?实际上,如果我们获得一个特定的值,举例说明.12.57,随机变量X的一个值,我们如何解释12.57发生的概率.It is interpreted as the probability of X assuming a value within a small interval around 12.57,i.e.12.565,12.575.解释为X假定一个值的概

25、率在一个小间距在12.57左右,也就是说12.565,12.575.This is obtained by integrating the area under the pdf between 12.565 and 12.575.在PDF下12.565 和 12.575之间的整个面积为此点的概率.P(X=x)=0for a continuousrandom variablePage 20Exponential DistributionArea of a line is zero!f(9.5)=P(X=9.5)=0To get probability of 20.0,integrate area

26、 between 19.995 and 20.005,i.e.P(19.995 X 10n)for inspection.让我们随机从一大批量样本(10n)中 取出 n个样本 Each part is classified asaccept or reject.每一部分被标识接受或拒收。Reject rate=pSample size(n)Page 28Binomial Experiment二项式实验二项式实验Assuming we have a process that is historically known to produce p reject rate.假设我们有一道工序，已知其历

27、史拒收率pp can be used as the probability of finding a failed unit each time we draw a part from the process for inspection.P用于当我们从工序每次取出一部用于当我们从工序每次取出一部分时，取到不合格品的概率。分时，取到不合格品的概率。Lets pull a sample of n partsrandomly from a large population(10n)for inspection.让我们随机从一大批量样本(10n)中 取出 n个样本 Each part is clas

28、sified asaccept or reject.每一部分被标识接受或拒收。For each trial(drawing a unit),the probability of success is constant.对于每次试验（取样本），成功的对于每次试验（取样本），成功的概率是一个常数概率是一个常数Trials are independent;result of a unit does not influence outcome of next unit试验是独立的，一个单位的结果不影试验是独立的，一个单位的结果不影响下一个结果的输出。响下一个结果的输出。Each trial resul

29、ts in only two possible outcomes.每一次试验只有两种可能的结果。每一次试验只有两种可能的结果。A binomial experiment!一个二项式试验一个二项式试验Page 29Probability Mass Function概率集中函数概率集中函数If each binomial experiment(pulling n parts randomly for pass/fail inspection)is repeated several times,do we see the same x defective units all the time?如果每

30、一个二项式实验（随机取n 个产品进行通过/拒收检查）被重复很多次，我们是否可以每次看到相同的X不合格品The pmf that describes how the x defective units(called successes)are distributed is given as:PMF描述X个不合格品（也叫合格品）的如何分布，表示为 n,0,1,2,xp1pxnxPxnxProbability of getting x defective units(x successes)得到得到X不合格品品不合格品品的概率（的概率（X合格品）合格品）Using a sample size of

31、n units(n trials)使用使用n个样本量（个样本量（n次）次）Given that the overall defective rate is p(probability of success is p)给出整个不合格品率给出整个不合格品率p(成功的概率是成功的概率是P）Page 30Applications应用应用The binomial distribution is extensively used to model results of experiments that generate binary outcomes,e.g.pass/fail,go/nogo,accep

32、t/reject,etc.二项式分布广泛应用于结果只输出两种的实验.举例来说,通过/不通过,去/不去,接受/拒绝.等等.In industrial practice,it is used for data generated from counting of defectives,e.g.:在工业实际中,常用于缺陷品计数的数据,举例来说1.Acceptance Sampling 接受样本2.p-chart P-ChartBinomial Distribution0.000.050.100.150.200.250.30012345678Number of Rejects(X)Probabilit

33、y of Finding X Rejects xnxp1pxnxPPage 31Example 1例例1If a process historically gives 10%reject rate(p=0.10),如果一个工序历史上拒绝率是10%(p=0.10),what is the chance of finding 0,1,2 or 3 defectives within a sample of 20 units(n=20)?则对于20个样本中发现0,1,2 或 3缺陷品的概略是多少?1.n,0,1,2,xp1pxnxPxnx.,0020101100200P0 xfor.1.011.01

34、201P,11021etcxforPage 32Example 1(contd)例例1继续继续These probabilities can be obtained from Minitab:这些概率可通过Minitab获得:Calc Probability Distributions BinomialP(x)n=20p=0.1包含X个缺陷品的指定列存储结果的指存储结果的指定列定列Page 33Example 1(contd)Binomial Distribution0.1220.2700.2850.1900.0900.0320.0090.0020.0000.000.050.100.150.2

35、00.250.30012345678No.of Defectives(x)Probability of Finding x Defectives n,0,1,2,xp1pxnxPxnxFrom Excel:From Minitab:What is the probability of getting 2 defectives or less?Page 34Example 1(contd)例例1(继续继续)For the 2 previous charts,the x-axis denotes the number of defective units,x.对于上页中的图表,X轴表明缺陷品单位的

36、数量 XIf we divide each x valueby constant sample size,n,and re-express the x-axisas a proportion defectivep-axis,the probabilitiesdo not change.如果我们将X除以恒定的样本量n,再重新代替X轴为缺陷品率p,则概率不变.Page 35The location,dispersion and shape of a binomial distribution are affected by the sample size,n,and defective rate,

37、p.二项式分布的位置,离散程度,和形状受样本量n和缺陷平率p影响.Parameters of Binomial Distribution二项式分布的参数二项式分布的参数分布参数Page 36Normal Approximation to the Binomial二项式分布的正态近似Depending on the values of n and p,the binomial distributions are a family of distributions that can be skewed to the left or right.依靠不同的n 和p,二项式分布是一个倾斜至左边或右边的

38、分布集合.Under certain conditions(combinations of n and p),the binomial distribution approximately approaches the shape of a normal distribution:在一定的情况下(n 和p一定),二项式分布近似于一个正态分布的形状.For p 0.5,np 5For p far from 0.5(smaller or larger),np 10Page 37Mean and Variance 均值和方差均值和方差Although n and p pin down a speci

39、fic binomial distribution,often the mean and variance of the distribution are used in practical applications such as the p-chart.尽管n 和 p 给定了一个特定的二项式分布,但分布的均值和方差经常被用于实际的分布,象p-chart.The mean and variance of a binomial distribution二项式分布的均值和方差ornppp12ppmpnnpppnnp12Page 38lBinomial Distribution 二项式分布lPoi

40、sson Distribution 泊松分布Page 39This distribution have been found to be relevant for applications involving error rates,particle count,chemical concentration,etc,此分布被发现应用于错误率,灰尘数,化学比,等等.where is the mean number of events(or defect rate)within a given unit of time or space.是给定的一个单位或空间中事件或缺陷率的平均数量.,2,1,0

41、 x!xexPxAnd where is small.Page 40Page 41Properties:unumber of outcomes in a time interval(or space region)is independent of the outcomes in another time interval(or space region)u单位时间(或空间)的数量输出独立于另一个单位时间(或空间)的数量输出.uprobability of an occurrence within a very short time interval(or space region)is pr

42、oportional to the time interval(or space region)u在非常短时间(或空间)内发生的概率是单位时间(或单位空间)输出数量的比率uprobability of more than 1 outcome occurring within a short time interval(or space region)is negligibleu极短时间(空间单位)内1个数量输出的概率可忽略不记uthe mean and variance for a Poisson Distribution areu泊松分布的均值和方差是2mandPage 42The loca

43、tion,dispersion and shape of a Poisson distribution is affected by the mean.泊松分布的位置,离散和形状都受均值影响Page 43A certain process yields a defect rate of 4 dpmo.For a million opportunities inspected,determine the probability distribution.某一工序产生的缺陷率是4dpmo.试计算其概率分布.Page 44Calc Probability Distributions Poissona

44、)Probability Mass Function b)Cumulative Distribution FunctionPage 45Binomial p 5if p 5np 10 if|p|Poisson Normal Page 46lNormal DistributionlExponential DistributionPage 47Normal DistributionPage 48The most widely used model for the distribution of continuous random variables.连续性随机变量应用最广泛的分布类型Arises

45、in the study of numerous natural physical phenomena,such as the velocity of molecules,as well as in one of the most important findings,the Central Limit Theorem.来自于大量自然物理现象的研究,例如分子的电压;中心极限定理也是许多非常重要发现的其中之一.Page 49Many natural phenomena and man-made processes are observed to have normal distributions

46、,or can be closely represented as normally distributed.我们观测到许多自然现象和人为工序都符合正态分布,或近似于正态分布.For example,the length of a machined part is observed to vary about its mean due to:例如:机器元件的长度均值的变化由于:temperature drift,humidity change,vibrations,cutting angle variations,cutting tool wear,bearing wear,rotationa

47、l speed variations,fixturing variations,raw material changes and contamination level changes温度漂移,湿度变化,振动,切削角度变化,切削工具磨损,轴承磨损,转速变化,夹具变化,原材料变更和污染级别变化,等等If these sources of variation are small,independent and equally likely to be positive or negative about the mean value,the length will closely approxim

48、ate a normal distribution.如果上述来源变化较小,独立和近似可能相对于均值偏正或偏负,则长度近似于一个正态分布.Page 50 dxxfxXPxFx)(Cumulative Distribution Function累计分布函数累计分布函数 xforexfx22121mNormal DistributionProbability Density Function概率密度函数概率密度函数aa0.5dxexx22121mPage 51A normal distribution can be completely described by knowing only the:一

49、个正态分布完全可以描述由已知的Mean(m)均值Variance(2)方差Distribution OneDistribution TwoDistribution ThreeWhat is the difference between the 3 normal distributions?三个正态分布有何不同三个正态分布有何不同?X N(m,2)1Parameters of the distribution分布分布2分布分布3分布分布1Page 52What is the difference between process A&B for each case?A,B 分布的区别?ANorma

50、l(m mA,A)BNormal(m mB,B)ANormal(m mA,A)BNormal(m mB,B)ANormal(m mA,A)BNormal(m mB,B)Page 53The mean,median and mode all coincide at the same value m.There is perfect symmetry.均值,中位数和重数一致为相同值 m,完全对称+-Does it mean that any data setwhich has mean,median and modeat the same value will automaticallybe a