工程方法_6 sigma_13.概率分布（凤山书屋）

1、Probability Distributions概率分布Page 1阳山书屋cLearning Objectives学习目的What is a Probability Distribution? 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Binomial Distribution 二项式分布Poisson Distribution 泊松分布.Hypergeom

2、etric distribution 超几何分布Continuous Distributions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Z Distribution Z 分布t Distribution t 分布c2 Distribution c2 分布F Distribution F 分布Pa

3、ge 2阳山书屋cAs we progress from description of 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

4、概念, 我们需要复习各种基本相关概念:Experiment, Sample Space, Event实验,样本空间,事件Random Variable 随机变量.What is a Probability Distribution?什么是概率分布?What do we mean by inference of data?Page 3阳山书屋cExperiment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

5、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 dataPinsStainsRejectAccept(b)Inspecting for stain marks检查污点印记Experiment generates attribute dataPins(c)Measuring shaft 测量 轴径10.53 mm10.49 mm10.22 mm10.29 mm11.20 mmExperiment generates continuous dataWhat is a Prob

6、ability Distribution?什么是概率分布?实验产生数字/离散数据实验产生计数性数据实验产生连续性数据Page 4阳山书屋cRandom Experiment 随机实验If we throw the dice 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

7、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 every 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What is a Probability Distribution?什么是概率分布?Page 5阳山书屋cSample Space样本空间The collection of all possible outcomes of

8、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An outcome, or a set of outcomes, from a random experiment is called an event, i.e. it is a subset 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What is a Probability Distribution?什么是概率分布?Page 6阳山书屋cEvent事件Example例 1: Some ev

9、ents 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 diameter 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

10、 = 5, 6 E1 = 1, 3, 5 E1= x mWhat is a Probability Distribution?什么是概率分布?Page 7阳山书屋cRandom Variable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, it is useful and convenient

11、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 shaft is within specs

12、.”.用W代表事件”轴径尺寸在规格内”What is a Probability Distribution?什么是概率分布?Page 8阳山书屋cRandom Variable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 c

13、alled a random variable because: 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 m

14、m,USL = 10 mm)0.,7.99998, 7.99999, 8, 8,00001,9.99999, 10, 10.00001, 10.00002, LSLUSLWhat is a Probability Distribution?什么是概率分布?Page 9阳山书屋cProbability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

15、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 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 rando

16、m 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 frequency 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What is a Probability Distribution?什么是概率分布?Page 10阳山书屋cProbability Distribution概率分布When we are able to assign a probability

17、 to each possible outcome of a random variable X, the full description of all the probabilities associated with the possible outcomes is 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:

18、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 axis每一个结果的概率在纵轴上What is a Probability Distribution?什么是概率分布?Page 11阳山书屋c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 关于概率Page 12阳山书屋c

19、Normal DistributionExponential DistributionUniform DistributionBinomial DistributionDiscrete Probability Distributions (Theoretical)离散概率分布(理论上)Continuous Probability Distributions (Theoretical)连续概率分布(理论上)What is a Probability Distribution?什么是概率分布?Page 13阳山书屋cEmpirical Distributions经验分布Created from a

20、ctual 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 14阳山书屋cThree common important characteristics:三个常用重要Shape- defines nature of distribution形

21、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Properties of Distributions分布的描述Exponential DistributionUniform Distribution统一分布指数分布Page 15阳山书屋cShape形状Describes how the probabilities of all the possible o

22、utcomes are distributed.描述所有可能结果可能性的分布Can be described mathematically with an equation called a probability function, e.g: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)Properties of Distributions分布的描述Pag

23、e 16阳山书屋c00f(t)1a2a3ab = 4210.5Probability Functions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Properties of Dis

24、tributions分布的描述Page 17阳山书屋cBinomial 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

25、(pdf), between two points.在一个连续分布,概率用曲线下两点间面积表示Properties of Distributions分布的描述Page 18阳山书屋cProbability of An Exact Value Under PDF is Zero!PDF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

26、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, 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 wit

27、hin a small interval around 12.57, i.e. 12.565, 12.575.解释为X假定一个值的概率在一个小间距在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 variableProperties of Distributions分布的描述Page 19阳山

28、书屋cExponential DistributionArea of a line is zero!f(9.5) = P(X = 9.5) = 0To get probability of 20.0, integrate area 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。Binomial Distribution二项式分布Reject rat

29、e = pSample size (n)Page 27阳山书屋cBinomial Experiment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分时，取到不合格品的概率。Let

30、s 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 classified asaccept or reject. 每一部分被标识接受或拒收。Binomial Distribution二项式分布For each trial (drawing a unit), the probability of success is constant.对于每次试验（取样本），成功的概率是一个常数Trials are ind

31、ependent; 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响下一个结果的输出。Each trial results in only two possible outcomes.每一次试验只有两种可能的结果。A binomial experiment!一个二项式试验Page 28阳山书屋cProbability Mass Function概率集中函数If each binomial experiment (pulling n parts randomly for pass/fail insp

32、ection) is repeated several times, do we see the same x defective units all the time?如果每一个二项式实验（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Probability of getting x defective uni

33、ts (x successes)得到X不合格品品的概率（X合格品）Using a sample size of n units (n trials)使用n个样本量（n次）Given that the overall defective rate is p(probability of success is p)给出整个不合格品率p(成功的概率是P） Binomial Distribution二项式分布Page 29阳山书屋cApplications应用The binomial distribution is extensively used to model results of experi

34、ments that generate binary outcomes, e.g. pass/fail, go/nogo, accept/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 Distrib

35、ution二项式分布Page 30阳山书屋cExample 1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.Binomial Distribution二项式分布Page 31阳山书屋cExample 1 (cont

36、d)例1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定列Binomial Distribution二项式分布Page 32阳山书屋cExample 1 (contd)From Excel:From Minitab:What is the probability of getting 2 defectives or less?Binomial Distribut

37、ion二项式分布Page 33阳山书屋cExample 1 (contd)例1(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

38、n,再重新代替X轴为缺陷品率p, 则概率不变.Binomial Distribution二项式分布Page 34阳山书屋cThe location, dispersion and shape of a binomial distribution are affected by the sample size, n, and defective rate, 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二项式分布的参数分布参数Binomial Distribution二项式分布Page 35阳山书屋cNor

39、mal 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分布集合.Under certain conditions (combinations of n and p), the binomial distribution approximately

40、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 10Binomial Distribution二项式分布Page 36阳山书屋cMean and Variance 均值和方差Although n and p pin down a specific binomial distribution, often the mean and variance of the distri

41、bution 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orBinomial Distribution二项式分布Page 37阳山书屋cImportantDiscrete Distributions重要的离散分布Binomial Distribution 二项式分布Poisson Distribution 泊松

42、分布Page 38阳山书屋cPoisson Distribution泊松分布This 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

43、缺陷率)的平均数量.And where is small.Page 39阳山书屋cSimeon D PoissonPage 40阳山书屋cPoisson Distribution泊松分布Properties:number of outcomes in a time interval (or space region) is independent of the outcomes in another time interval (or space region)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.probability of an occurrence wit

44、hin a very short time interval (or space region) is proportional to the time interval (or space region)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probability of more than 1 outcome occurring within a short time interval (or space region) is negligible极短时间(空间单位)内1个数量输出的概率可忽略不记the mean and variance for a Poi

45、sson Distribution are泊松分布的均值和方差是andPage 41阳山书屋cPoisson Distribution泊松分布The location, 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 42阳山书屋cExample 2练习2.A certain process yields a defect rate of 4 dpmo. For a million opportunities inspected, determine t

46、he probability distribution.某一工序产生的缺陷率是4dpmo. 试计算其概率分布.Page 43阳山书屋cExample 2Calc Probability Distributions Poissona) Probability Mass Function b) Cumulative Distribution FunctionPage 44阳山书屋cSummary of Approximations近似总结Binomial p 5if p 5np 10 if |p| Poisson Normal Page 45阳山书屋cImportantContinuous Dis

47、tributionsNormal DistributionExponential DistributionPage 46阳山书屋cNormal Distribution正态分布Normal DistributionPage 47阳山书屋cThe 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 in the study of numerous natural physical phenomena, such as the velocity of m

48、olecules, 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.Normal Distribution正态分布Page 48阳山书屋cMany natural phenomena and man-made processes are observed to have normal distributions, or can be closely represented as normally d

49、istributed.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, rotational speed variations, fixturing variat

50、ions, 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 approximate 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.Normal Distribution正态分布Page 49阳山书屋cCumulative Distribution Functio