工程方法_6 sigma_13.概率分布

1、整理整理pptPage 1整理整理pptPage 2What is a Probability Distribution? 什么是概率分布什么是概率分布?Experiment, Sample Space, Event 實驗,樣本空間,事件Random Variable, Probability Functions (pmf, pdf, cdf)隨機變量,概率函數(shù)Discrete Distributions離散分布離散分布Binomial Distribution 二項式分布Poisson Distribution 泊松分布.Hypergeometric distribution 超幾何分布Co

2、ntinuous Distributions連續(xù)分布連續(xù)分布Normal Distribution 正態(tài)分布Uniform distribution 均勻分布Exponential distribution 指數(shù)分布Logarithmic normal distribution 對數(shù)正態(tài)分布Weibull distribution 威布爾分布Sampling Distributions樣本分布樣本分布Z Distribution Z 分布t Distribution t 分布c2 Distribution c2 分布F Distribution F 分布整理整理pptPage 3As we p

3、rogress from description of data towards inference of data, an important concept is the idea of a probability distribution.當(dāng)我們從描述性數(shù)據(jù)進步到推論性數(shù)據(jù)時,一個重要的內(nèi)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:為了解概率分布的概念, 我們需要復(fù)習(xí)各種基本相關(guān)

4、概念:Experiment, Sample Space, Event實驗,樣本空間,事件Random Variable 隨機變量.What do we mean by inference of data?整理整理pptPage 4Experiment實驗實驗An experiment is any activity that generates a set of data, which may be numerical or not numerical.實驗是產(chǎn)生一系列數(shù)據(jù)的行為,數(shù)據(jù)有可能是數(shù)字的或非數(shù)字的.1, 2, ., 6(a)Throwing a dice擲骰子Experiment

5、generates numerical / discrete dataPinsStainsRejectAccept(b)Inspecting for stain marks檢檢查污點印記查污點印記Experiment generates attribute dataPins(c)Measuring shaft 測量 軸徑10.53 mm10.49 mm10.22 mm10.29 mm11.20 mmExperiment generates continuous data實驗產(chǎn)生數(shù)字實驗產(chǎn)生數(shù)字/離散數(shù)據(jù)離散數(shù)據(jù)實驗產(chǎn)生計數(shù)性數(shù)據(jù)實驗產(chǎn)生計數(shù)性數(shù)據(jù)實驗產(chǎn)生連續(xù)性數(shù)據(jù)實驗產(chǎn)生連續(xù)性數(shù)據(jù)整理整理pp

6、tPage 5Random Experiment 隨機實驗隨機實驗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.如果我們擲子一次由一次,或從相同工序生產(chǎn)許多軸,結(jié)果會是不同的.不能完全提前預(yù)測.An experiment which can result in different

7、outcomes, even though it is repeated in the same manner every time, is called a random experiment.一個實驗導(dǎo)致不同的結(jié)果,即使它是每次以相同方式,這叫做隨機實驗整理整理pptPage 6Sample Space樣本空間樣本空間The collection of all possible outcomes of an experiment is called its sample space.收集實驗的所有可能結(jié)果稱為樣本空間Event事件事件An outcome, or a set of outc

8、omes, from a random experiment is called an event, i.e. it is a subset of the sample space.一個結(jié)果,或一套結(jié)果,從一個隨機實驗出來的稱為事件,也就是樣本空間的子集整理整理pptPage 7Event事件事件Example例 1: Some events from tossing of a dice.從擲骰子的一些事件.Event 事件1: the outcome is an odd number 結(jié)果是奇數(shù)Event事件 2: the outcome is a number 4 大于4的結(jié)果Exampl

9、e例 2: Some events from measuring shaft :從測量軸徑的一些事件Event事件 1: the outcome is a diameter mean直徑大于平均值Event 事件2: the outcome is a part failing specs.未通過規(guī)格的結(jié)果. E2 = x USL E2 = 5, 6 E1 = 1, 3, 5 E1= x m整理整理pptPage 8Random Variable隨機變量隨機變量From a same experiment, different events can be derived depending on

10、 which aspects of the experiment we consider important.從一個相同的實驗, 由于我們認為重要的實驗方面不同而產(chǎn)生不同的結(jié)果In many cases, it 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.: 許多方面,它是很有用和方便的定義我們感興趣的實驗

11、方面, 通過一個大寫的字母表示.舉例說明:Let X be the event “the number of a dice is odd”.用X代表事件”骰子的數(shù)字是奇數(shù)”Let W be the event “the shaft is within specs.”.用W代表事件”軸徑尺寸在規(guī)格內(nèi)”整理整理pptPage 9Random Variable隨機變量隨機變量We have defined a function that assigns a real number to an experimental outcome within the sample space of the ra

12、ndom experiment.我們定義了一個函數(shù),其代表了一個在隨機實驗的樣本空間的一個真實實驗數(shù)字This function (X or W in our examples) is called a random variable because: 函數(shù)(例子中的X 或W )稱為隨機變量,是因為:The outcomes of the same event are clearly uncertain and are variable from one outcome to another一個事件的發(fā)生結(jié)果是明顯不定的,是同另一個結(jié)果相異的.Each outcome has an equal

13、 chance of being selected.每一個結(jié)果有相同被選擇的機會.PinsMeasuring shaft X = Parts out of specs.(LSL = 8 mm,USL = 10 mm)0.,7.99998, 7.99999, 8, 8,00001,9.99999, 10, 10.00001, 10.00002, LSLUSL整理整理pptPage 10Probability概率概率To quantify how likely a particular outcome of a random variable can occur, we typically ass

14、ign a numerical value between 0 and 1 (or 0 to 100%).為量化一個隨機變量的指定結(jié)果發(fā)生的可能性,我們指定一個數(shù)字介于0和1之間(或0100%)This numerical value is called the probability of the outcome.這個數(shù)字稱為結(jié)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

15、the outcome occurs in many repetitions of the same random experiment.有幾種方式解釋概率.一般的方式是解釋概率為在許多相同實驗重復(fù)后發(fā)生的分數(shù)(或比例)次數(shù)This method is the relative frequency approach or frequentist approach to interpreting probability.這種方法概率解釋的相對頻率模擬或單位頻率模擬整理整理pptPage 11Probability Distribution概率分布概率分布When we are able to a

16、ssign a probability 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.當(dāng)我們能夠表明一個隨機變量的某一個可能結(jié)果的概率,則整個可能結(jié)果的概率的描述稱為X的概率分布A probability distribution is typically presented as a curv

17、e or plot that has:一個概率分布被代表為一個曲線或點應(yīng)有:All the possible outcomes of X on the horizontal axisX的所有的可能結(jié)果在水平軸線上The probability of each outcome on the vertical axis每一個結(jié)果的概率在縱軸上整理整理pptPage 12隨機現(xiàn)象 隨機試驗 樣本點、樣本空間 語言表示 事件的表示 集合表示 事件的特征 包含、相等 隨機事件 事件間的關(guān)系 互斥 事件的運算： 對立、并、交、差 整理整理pptPage 13Normal DistributionExpon

18、ential DistributionUniform DistributionBinomial DistributionDiscrete Probability Distributions (Theoretical)離散概率分布離散概率分布(理論上理論上)Continuous Probability Distributions (Theoretical)連續(xù)概率分布連續(xù)概率分布(理論上理論上)整理整理pptPage 14Created from actual observations. Usually represented as histograms.根據(jù)實際觀測得來, 通常用直方圖代表Em

19、pirical distributions, like theoretical distributions, apply to both discrete and continuous distributions.經(jīng)驗分布,象理論上的分布,適用于離散和連續(xù)分布.整理整理pptPage 15Three common important characteristics:三個常用重要Shape- defines nature of distribution形狀 - 定義分布的自然性Center- defines central tendency of data中心 - 定義中心趨勢的數(shù)據(jù)Spread

20、分布(或離散,或刻度)- defines dispersion of data(or Dispersion, or Scale) 定義數(shù)據(jù)的離散Exponential DistributionUniform Distribution統(tǒng)一分布統(tǒng)一分布指數(shù)分布指數(shù)分布整理整理pptPage 16 xexfx22121mShape形狀形狀lDescribes how the probabilities of all the possible outcomes are distributed.l描述所有可能結(jié)果可能性的分布lCan be described mathematically with an

21、 equation called a probability function, e.g:l可以用一個概率函數(shù)數(shù)字表示,舉例說明Probability function概率函數(shù)Lowercase letter represents a specific value of random variable X小字母代表隨機變量X某一個特定值 f(x) means P(X = x)整理整理pptPage 1700f(t)1a2a3ab = 4210.5Probability Functions概率函數(shù)概率函數(shù)For a discrete distribution,對于一個離散分布f(x) calle

22、d is the probability f(x) 稱為概率集中:mass function (pmf), e.g.:函數(shù),舉例說明For a continuous distribution,對于一個連續(xù)分布f(x) is called the probability f(x) 稱為概率密度density function (pdf), e.g.:函數(shù)舉例說明 n,0,1,2,xp1pxnxPxnx 0,1tettftbabbab整理整理pptPage 18Binomial DistributionNormal DistributionThe total probability for any

23、 distribution sums to 1.任何分布的全部概率總和為1In a discrete distribution,probability is representedas height of the bar.在一個離散分布,概率用柱狀表示In a continuous distribution,probability is representedas area under the curve(pdf), between two points.在一個連續(xù)分布,概率用曲線下兩點間面積表示整理整理pptPage 19Probability of An Exact Value Under

24、 PDF is Zero!PDF下一個準確值的概率是零下一個準確值的概率是零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:對于一個連續(xù)隨機變量,一個準確值發(fā)生的概率理論上是0,是因為PDF上一條線的寬度是0”.意味著:In practice, if we obtain a particular value, e.g. 12.57, of a random v

25、ariable X, how do we interpret the probability of 12.57 happening?實際上,如果我們獲得一個特定的值,舉例說明.12.57, 隨機變量X的一個值, 我們?nèi)绾谓忉?2.57發(fā)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.解釋為X假定一個值的概率在一個小間距在12.57左右,也就是說12.565, 12.575.This is obtain

26、ed by integrating the area under the pdf between 12.565 and 12.575.在PDF下12.565 和 12.575之間的整個面積為此點的概率.P(X = x) = 0for a continuousrandom variable整理整理pptPage 20Exponential 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.

27、e.P(19.995 X 10n) for inspection. 讓我們隨機從一大批量樣本( 10n)中 取出 n個樣本 Each part is classified asaccept or reject. 每一部分被標識接受或拒收。Reject rate = pSample size (n)整理整理pptPage 28Binomial Experiment二項式實驗二項式實驗Assuming we have a process that is historically known to produce p reject rate.假設(shè)我們有一道工序，已知其歷史拒收率pp can be u

28、sed as the probability of finding a failed unit each time we draw a part from the process for inspection.P用于當(dāng)我們從工序每次取出一部用于當(dāng)我們從工序每次取出一部分時，取到不合格品的概率。分時，取到不合格品的概率。Lets pull a sample of n partsrandomly from a large population( 10n) for inspection. 讓我們隨機從一大批量樣本( 10n)中 取出 n個樣本 Each part is classified asac

29、cept or reject. 每一部分被標識接受或拒收。For each trial (drawing a unit), the probability of success is constant.對于每次試驗（取樣本），成功的對于每次試驗（取樣本），成功的概率是一個常數(shù)概率是一個常數(shù)Trials are independent; result of a unit does not influence outcome of next unit試驗是獨立的，一個單位的結(jié)果不影試驗是獨立的，一個單位的結(jié)果不影響下一個結(jié)果的輸出。響下一個結(jié)果的輸出。Each trial results in o

30、nly two possible outcomes.每一次試驗只有兩種可能的結(jié)果。每一次試驗只有兩種可能的結(jié)果。A binomial experiment!一個二項式試驗一個二項式試驗整理整理pptPage 29Probability Mass Function概率集中函數(shù)概率集中函數(shù)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?

31、如果每一個二項式實驗（隨機取n 個產(chǎn)品進行通過/拒收檢查）被重復(fù)很多次，我們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個不合格品（也叫合格品）的如何分布，表示為 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 s

32、ize of n units (n trials)使用使用n個樣本量（個樣本量（n次）次）Given that the overall defective rate is p(probability of success is p)給出整個不合格品率給出整個不合格品率p(成功的概率是成功的概率是P） 整理整理pptPage 30Applications應(yīng)用應(yīng)用The binomial distribution is extensively used to model results of experiments that generate binary outcomes, e.g. pass/

33、fail, go/nogo, accept/reject, etc.二項式分布廣泛應(yīng)用于結(jié)果只輸出兩種的實驗.舉例來說,通過/不通過,去/不去,接受/拒絕.等等.In industrial practice, it is used for data generated from counting of defectives, e.g.:在工業(yè)實際中,常用于缺陷品計數(shù)的數(shù)據(jù),舉例來說1. Acceptance Sampling 接受樣本2. p-chart P-ChartBinomial Distribution0.000.050.100.150.200.250.30012345678Numbe

34、r of Rejects (X)Probability of Finding X Rejects xnxp1pxnxP整理整理pptPage 31Example 1例例1If a process historically gives 10% reject rate (p = 0.10), 如果一個工序歷史上拒絕率是10% (p = 0.10), what is the chance of finding 0, 1, 2 or 3 defectives within a sample of 20 units (n = 20)?則對于20個樣本中發(fā)現(xiàn)0, 1, 2 或 3缺陷品的概略是多少?1.

35、n ,0,1,2,xp1pxnxPxnx .,0020101100200P0 xfor .1 . 011 . 01201P, 11021etcxfor整理整理pptPage 32Example 1 (contd)例例1繼續(xù)繼續(xù)These probabilities can be obtained from Minitab:這些概率可通過Minitab獲得:Calc Probability Distributions BinomialP(x)n = 20p = 0.1包含X個缺陷品的指定列存儲結(jié)果的指定存儲結(jié)果的指定列列整理整理pptPage 33Example 1 (contd)Binomia

36、l Distribution0.1220.2700.2850.1900.0900.0320.0090.0020.0000.000.050.100.150.200.250.30012345678No. 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?整理整理pptPage 34Example 1 (contd)例例1(繼續(xù)繼續(xù))For the

37、 2 previous charts, the x-axis denotes the number of defective units, x.對于上頁中的圖表,X軸表明缺陷品單位的數(shù)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.如果我們將X除以恒定的樣本量n,再重新代替X軸為缺陷品率p, 則概率不變.整理整理pptPage 35The location,

38、 dispersion and shape of a binomial distribution are affected by the sample size, n, and defective rate, p.二項式分布的位置,離散程度,和形狀受樣本量n和缺陷平率p影響.Parameters of Binomial Distribution二項式分布的參數(shù)二項式分布的參數(shù)分布參數(shù)整理整理pptPage 36Normal Approximation to the Binomial二項式分布的正態(tài)近似Depending on the values of n and p, the binomia

39、l distributions are a family of distributions that can be skewed to the left or right.依靠不同的n 和p,二項式分布是一個傾斜至左邊或右邊的分布集合.Under certain conditions (combinations of n and p), the binomial distribution approximately approaches the shape of a normal distribution:在一定的情況下(n 和p一定),二項式分布近似于一個正態(tài)分布的形狀.For p 0.5,

40、np 5For p far from 0.5 (smaller or larger),np 10整理整理pptPage 37Mean and Variance 均值和方差均值和方差A(yù)lthough n and p pin down a specific binomial distribution, often the mean and variance of the distribution are used in practical applications such as the p-chart.盡管n 和 p 給定了一個特定的二項式分布,但分布的均值和方差經(jīng)常被用于實際的分布,象p-ch

41、art.The mean and variance of a binomial distribution二項式分布的均值和方差ornppp12ppmpnnpppnnp12整理整理pptPage 38lBinomial Distribution 二項式分布lPoisson Distribution 泊松分布整理整理pptPage 39This distribution have been found to be relevant for applications involving error rates, particle count, chemical concentration, etc,

42、此分布被發(fā)現(xiàn)應(yīng)用于錯誤率,灰塵數(shù),化學(xué)比,等等.where is the mean number of events (or defect rate) within a given unit of time or space.是給定的一個單位或空間中事件或缺陷率的平均數(shù)量. , 2 , 1 , 0 x!xexPxAnd where is small.整理整理pptPage 40整理整理pptPage 41Properties:unumber of outcomes in a time interval (or space region) is independent of the outcom

43、es in another time interval (or space region)u單位時間(或空間)的數(shù)量輸出獨立于另一個單位時間(或空間)的數(shù)量輸出.uprobability of an occurrence within a very short time interval (or space region) is proportional to the time interval (or space region)u在非常短時間(或空間)內(nèi)發(fā)生的概率是單位時間(或單位空間)輸出數(shù)量的比率uprobability of more than 1 outcome occurring

44、within a short time interval (or space region) is negligibleu極短時間(空間單位)內(nèi)1個數(shù)量輸出的概率可忽略不記uthe mean and variance for a Poisson Distribution areu泊松分布的均值和方差是2mand整理整理pptPage 42The location, dispersion and shape of a Poisson distribution is affected by the mean.泊松分布的位置,離散和形狀都受均值影響整理整理pptPage 43A certain pr

45、ocess yields a defect rate of 4 dpmo. For a million opportunities inspected, determine the probability distribution.某一工序產(chǎn)生的缺陷率是4dpmo. 試計算其概率分布.整理整理pptPage 44Calc Probability Distributions Poissona) Probability Mass Function b) Cumulative Distribution Function整理整理pptPage 45Binomial p 5if p 5np 10 if

46、|p| Poisson Normal 整理整理pptPage 46lNormal DistributionlExponential Distribution整理整理pptPage 47Normal Distribution整理整理pptPage 48The most widely used model for the distribution of continuous random variables.連續(xù)性隨機變量應(yīng)用最廣泛的分布類型Arises in the study of numerous natural physical phenomena, such as the velocit

47、y of molecules, as well as in one of the most important findings, the Central Limit Theorem.來自于大量自然物理現(xiàn)象的研究, 例如分子的電壓; 中心極限定理也是許多非常重要發(fā)現(xiàn)的其中之一.整理整理pptPage 49Many natural phenomena and man-made processes are observed to have normal distributions, or can be closely represented as normally distributed.我們觀測

48、到許多自然現(xiàn)象和人為工序都符合正態(tài)分布,或近似于正態(tài)分布.For example, the length of a machined part is observed to vary about its mean due to:例如: 機器元件的長度均值的變化由于:temperature drift, humidity change, vibrations, cutting angle variations, cutting tool wear, bearing wear, rotational speed variations, fixturing variations, raw mater

49、ial changes and contamination level changes溫度漂移,濕度變化,振動,切削角度變化,切削工具磨損,軸承磨損,轉(zhuǎn)速變化,夾具變化,原材料變更和污染級別變化,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.如果上述來源變化較小,獨立和近似可能相對于

50、均值偏正或偏負,則長度近似于一個正態(tài)分布.整理整理pptPage 50 dxxfxXPxFx)(Cumulative Distribution Function累計分布函數(shù)累計分布函數(shù) xforexfx22121mNormal DistributionProbability Density Function概率密度函數(shù)概率密度函數(shù)aa0.5dxexx22121m整理整理pptPage 51A normal distribution can be completely described by knowing only the:一個正態(tài)分布完全可以描述由已知的Mean (m)均值Variance

51、 (2)方差Distribution OneDistribution TwoDistribution ThreeWhat is the difference between the 3 normal distributions?三個正態(tài)分布有何不同三個正態(tài)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1整理整理pptPage 52What is the difference between process A & B for each case?A,B 分布的區(qū)別?ANormal(m mA, A)BNormal

52、(m mB, B)ANormal(m mA, A)BNormal(m mB, B)ANormal(m mA, A)BNormal(m mB, B)整理整理pptPage 53The mean, median and mode all coincide at the same value m. There is perfect symmetry.均值,中位數(shù)和重數(shù)一致為相同值 m, 完全對稱+ - Does it mean that any data setwhich has mean, median and modeat the same value will automaticallybe

53、a normal distribution?是否上述三個參數(shù)一致的分布就是正態(tài)分布?MeanMedianMode2整理整理pptPage 54The area under sections of the curve can be used to estimate the probability of a certain “event” occurring:部分曲線下的面積可用于計算一定事件發(fā)生的概率Point of Inflection1 + - 68.27%95.45%99.73%m+/- 3 is often referred to as the width of a normal dis

54、tribution(常指正態(tài)分布的寬度)3整理整理pptPage 55Lets compute the cumulative probabilities of the following distributions:讓我們計算下列分布的累計概率+ - m = 3.5 = 0.61.8+ - 20.0m = 16.6 = 2.8+ - m = -1.5 = 0.9-2.80.5F(1.8) = P(X 20.0) = 1 F(20.0)?P(-2.8 X 2T =x ms /nN =80(N0)-40.0187240.0005247160.000247009-3.90.0196370.00068

55、63350.000345186-3.80.0206160.0008957220.000479601-3.70.0216680.0011661360.000662401-3.60.0228020.0015141710.000909295-3.50.0240230.0019604560.001240393-3.40.0253430.0025304640.001681176-3.30.0267710.0032554110.002263577-3.20.0283190.0041732320.003027153-3.10.0300010.0053296170.004020315-30.0318310.0

56、067790630.005301541-2.90.0338270.008585870.006940505-2.80.0360080.0108250170.009018995-2.70.0383970.0135827940.011631492-2.60.0410190.0169570680.014885242-2.50.0439050.0210570190.018899647-2.40.0470870.0260021740.023804777-2.30.0506060.0319205490.029738823-2.20.0545050.0389457250.036844331-2.10.0588

57、370.0472126780.045263115-20.0636620.0568522750.055129788-1.90.0690480.0679843770.066563991-1.80.0750730.0807096250.079661505-1.70.0818280.0951001110.094484539-1.60.0894130.1111892810.111051695-1.50.0979420.1289616020.129328186-1.40.1075370.1483426790.149217047-1.30.1183310.1691906490.170552117-1.20.

58、1304550.1912897650.193093619-1.10.1440320.2143471180.216527085Input Number of Degrees of Freedom in RED :t Distribution00.10.20.30.4-4-3-2-101234Tv = 1v = 30v = NInteractive SlideCompare against Z, we have replaced using s in the T statistic.整理整理pptPage 66故有一個 t 分布其自由度 n = n -1E(T) = 0V(T) =nn 2, n

59、2T =x ms /nN =80(N0)-40.0187240.0005247160.000247009-3.90.0196370.0006863350.000345186-3.80.0206160.0008957220.000479601-3.70.0216680.0011661360.000662401-3.60.0228020.0015141710.000909295-3.50.0240230.0019604560.001240393-3.40.0253430.0025304640.001681176-3.30.0267710.0032554110.002263577-3.20.0283

60、190.0041732320.003027153-3.10.0300010.0053296170.004020315-30.0318310.0067790630.005301541-2.90.0338270.008585870.006940505-2.80.0360080.0108250170.009018995-2.70.0383970.0135827940.011631492-2.60.0410190.0169570680.014885242-2.50.0439050.0210570190.018899647-2.40.0470870.0260021740.023804777-2.30.05060