1、Probability Distributions概率分布Page 1陽山書屋cLearning Objectives學(xué)習(xí)目的What is a Probability Distribution? 什么是概率分布?Experiment, Sample Space, Event 實(shí)驗(yàn),樣本空間,事件Random Variable, Probability Functions (pmf, pdf, cdf)隨機(jī)變量,概率函數(shù)Discrete Distributions離散分布Binomial Distribution 二項(xiàng)式分布Poisson Distribution 泊松分布.Hypergeom

2、etric distribution 超幾何分布Continuous Distributions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(duì)數(shù)正態(tài)分布Weibull distribution 威布爾分布Sampling Distributions樣本分布Z Distribution Z 分布t Distribution t 分布c2 Distribution c2 分布F Distribution F 分布Pa

3、ge 2陽山書屋cAs we progress 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(jìn)步到推論性數(shù)據(jù)時(shí),一個(gè)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

4、概念, 我們需要復(fù)習(xí)各種基本相關(guān)概念:Experiment, Sample Space, Event實(shí)驗(yàn),樣本空間,事件Random Variable 隨機(jī)變量.What is a Probability Distribution?什么是概率分布?What do we mean by inference of data?Page 3陽山書屋cExperiment實(shí)驗(yàn)An experiment is any activity that generates a set of data, which may be numerical or not numerical.實(shí)驗(yàn)是產(chǎn)生一系列數(shù)據(jù)的行為,數(shù)據(jù)

5、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 generates numerical / discrete dataPinsStainsRejectAccept(b)Inspecting for stain marks檢查污點(diǎn)印記Experiment generates attribute dataPins(c)Measuring shaft 測(cè)量 軸徑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?實(shí)驗(yàn)產(chǎn)生數(shù)字/離散數(shù)據(jù)實(shí)驗(yàn)產(chǎn)生計(jì)數(shù)性數(shù)據(jù)實(shí)驗(yàn)產(chǎn)生連續(xù)性數(shù)據(jù)Page 4陽山書屋cRandom Experiment 隨機(jī)實(shí)驗(yàn)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(huì)

7、是不同的.不能完全提前預(yù)測(cè).An experiment which can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment.一個(gè)實(shí)驗(yàn)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(jī)實(shí)驗(yàn)What is a Probability Distribution?什么是概率分布?Page 5陽山書屋cSample Space樣本空間The collection of all possible outcomes of

8、an experiment is called its sample space.收集實(shí)驗(yàn)的所有可能結(jié)果稱為樣本空間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.一個(gè)結(jié)果,或一套結(jié)果,從一個(gè)隨機(jī)實(shí)驗(yàn)出來的稱為事件,也就是樣本空間的子集What is a Probability Distribution?什么是概率分布?Page 6陽山書屋cEvent事件Example例 1: Some ev

9、ents 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Example例 2: Some events from measuring shaft :從測(cè)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

10、 = 5, 6 E1 = 1, 3, 5 E1= x mWhat is a Probability Distribution?什么是概率分布?Page 7陽山書屋cRandom Variable隨機(jī)變量From a same experiment, different events can be derived depending on which aspects of the experiment we consider important.從一個(gè)相同的實(shí)驗(yàn), 由于我們認(rèn)為重要的實(shí)驗(yàn)方面不同而產(chǎn)生不同的結(jié)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.: 許多方面,它是很有用和方便的定義我們感興趣的實(shí)驗(yàn)方面, 通過一個(gè)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

12、.”.用W代表事件”軸徑尺寸在規(guī)格內(nèi)”What is a Probability Distribution?什么是概率分布?Page 8陽山書屋cRandom Variable隨機(jī)變量We have defined a function that assigns a real number to an experimental outcome within the sample space of the random experiment.我們定義了一個(gè)函數(shù),其代表了一個(gè)在隨機(jī)實(shí)驗(yàn)的樣本空間的一個(gè)真實(shí)實(shí)驗(yàn)數(shù)字This function (X or W in our examples) is c

13、alled a random variable because: 函數(shù)(例子中的X 或W )稱為隨機(jī)變量,是因?yàn)?The outcomes of the same event are clearly uncertain and are variable from one outcome to another一個(gè)事件的發(fā)生結(jié)果是明顯不定的,是同另一個(gè)結(jié)果相異的.Each outcome has an equal chance of being selected.每一個(gè)結(jié)果有相同被選擇的機(jī)會(huì).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陽山書屋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%).為量化

15、一個(gè)隨機(jī)變量的指定結(jié)果發(fā)生的可能性,我們指定一個(gè)數(shù)字介于0和1之間(或0100%)This numerical value is called the probability of the outcome.這個(gè)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 the outcome occurs in many repetitions of the same rando

16、m 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(shí)驗(yàn)重復(fù)后發(fā)生的分?jǐn)?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(duì)頻率模擬或單位頻率模擬What is a Probability Distribution?什么是概率分布?Page 10陽山書屋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.當(dāng)我們能夠表明一個(gè)隨機(jī)變量的某一個(gè)可能結(jié)果的概率,則整個(gè)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 curve or plot that has:

18、一個(gè)概率分布被代表為一個(gè)曲線或點(diǎn)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(gè)結(jié)果的概率在縱軸上What is a Probability Distribution?什么是概率分布?Page 11陽山書屋c隨機(jī)現(xiàn)象 隨機(jī)試驗(yàn) 樣本點(diǎn)、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(jī)事件 事件間的關(guān)系 互斥 事件的運(yùn)算： 對(duì)立、并、交、差 關(guān)于概率Page 12陽山書屋c

19、Normal DistributionExponential DistributionUniform DistributionBinomial DistributionDiscrete Probability Distributions (Theoretical)離散概率分布(理論上)Continuous Probability Distributions (Theoretical)連續(xù)概率分布(理論上)What is a Probability Distribution?什么是概率分布?Page 13陽山書屋cEmpirical Distributions經(jīng)驗(yàn)分布Created from a

20、ctual observations. Usually represented as histograms.根據(jù)實(shí)際觀測(cè)得來, 通常用直方圖代表Empirical distributions, like theoretical distributions, apply to both discrete and continuous distributions.經(jīng)驗(yàn)分布,象理論上的分布,適用于離散和連續(xù)分布.Page 14陽山書屋cThree common important characteristics:三個(gè)常用重要Shape- defines nature of distribution形

21、狀 - 定義分布的自然性Center- defines central tendency of data中心 - 定義中心趨勢(shì)的數(shù)據(jù)Spread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Properties of Distributions分布的描述Exponential DistributionUniform Distribution統(tǒng)一分布指數(shù)分布Page 15陽山書屋cShape形狀Describes how the probabilities of all the possible o

22、utcomes are distributed.描述所有可能結(jié)果可能性的分布Can be described mathematically with an equation called a probability function, e.g:可以用一個(gè)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(jī)變量X某一個(gè)特定值 f(x) means P(X = x)Properties of Distributions分布的描述Pag

23、e 16陽山書屋c00f(t)1a2a3ab = 4210.5Probability Functions概率函數(shù)For a discrete distribution,對(duì)于一個(gè)離散分布f(x) called 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(duì)于一個(gè)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Properties of Dis

24、tributions分布的描述Page 17陽山書屋cBinomial DistributionNormal DistributionThe total probability for any 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(gè)離散分布,概率用柱狀表示In a continuous distribution,probability is representedas area under the curve

25、(pdf), between two points.在一個(gè)連續(xù)分布,概率用曲線下兩點(diǎn)間面積表示Properties of Distributions分布的描述Page 18陽山書屋cProbability of An Exact Value Under PDF is Zero!PDF下一個(gè)準(zhǔn)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(duì)于一個(gè)

26、連續(xù)隨機(jī)變量,一個(gè)準(zhǔn)確值發(fā)生的概率理論上是0,是因?yàn)镻DF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 variable X, how do we interpret the probability of 12.57 happening?實(shí)際上,如果我們獲得一個(gè)特定的值,舉例說明.12.57, 隨機(jī)變量X的一個(gè)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 wit

27、hin a small interval around 12.57, i.e. 12.565, 12.575.解釋為X假定一個(gè)值的概率在一個(gè)小間距在12.57左右,也就是說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之間的整個(gè)面積為此點(diǎn)的概率.P(X = x) = 0for a continuousrandom variableProperties of Distributions分布的描述Page 19陽山

28、書屋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. 讓我們隨機(jī)從一大批量樣本( 10n)中 取出 n個(gè)樣本 Each part is classified asaccept or reject. 每一部分被標(biāo)識(shí)接受或拒收。Binomial Distribution二項(xiàng)式分布Reject rat

29、e = pSample size (n)Page 27陽山書屋cBinomial Experiment二項(xiàng)式實(shí)驗(yàn)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 used 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分時(shí)，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. 讓我們隨機(jī)從一大批量樣本( 10n)中 取出 n個(gè)樣本 Each part is classified asaccept or reject. 每一部分被標(biāo)識(shí)接受或拒收。Binomial Distribution二項(xiàng)式分布For each trial (drawing a unit), the probability of success is constant.對(duì)于每次試驗(yàn)（取樣本），成功的概率是一個(gè)常數(shù)Trials are ind

31、ependent; result of a unit does not influence outcome of next unit試驗(yàn)是獨(dú)立的，一個(gè)單位的結(jié)果不影響下一個(gè)結(jié)果的輸出。Each trial results in only two possible outcomes.每一次試驗(yàn)只有兩種可能的結(jié)果。A binomial experiment!一個(gè)二項(xiàng)式試驗(yàn)Page 28陽山書屋cProbability Mass Function概率集中函數(shù)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?如果每一個(gè)二項(xiàng)式實(shí)驗(yàn)（隨機(jī)取n 個(gè)產(chǎn)品進(jìn)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(gè)不合格品（也叫合格品）的如何分布，表示為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個(gè)樣本量（n次）Given that the overall defective rate is p(probability of success is p)給出整個(gè)不合格品率p(成功的概率是P） Binomial Distribution二項(xiàng)式分布Page 29陽山書屋cApplications應(yīng)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.二項(xiàng)式分布廣泛應(yīng)用于結(jié)果只輸出兩種的實(shí)驗(yàn).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(shí)際中,常用于缺陷品計(jì)數(shù)的數(shù)據(jù),舉例來說1. Acceptance Sampling 接受樣本2. p-chart P-ChartBinomial Distrib

35、ution二項(xiàng)式分布Page 30陽山書屋cExample 1例1If a process historically gives 10% reject rate (p = 0.10), 如果一個(gè)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(duì)于20個(gè)樣本中發(fā)現(xiàn)0, 1, 2 或 3缺陷品的概略是多少?1.Binomial Distribution二項(xiàng)式分布Page 31陽山書屋cExample 1 (cont

36、d)例1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(gè)缺陷品的指定列存儲(chǔ)結(jié)果的指定列Binomial Distribution二項(xiàng)式分布Page 32陽山書屋cExample 1 (contd)From Excel:From Minitab:What is the probability of getting 2 defectives or less?Binomial Distribut

37、ion二項(xiàng)式分布Page 33陽山書屋cExample 1 (contd)例1(繼續(xù))For the 2 previous charts, the x-axis denotes the number of defective units, x.對(duì)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

38、n,再重新代替X軸為缺陷品率p, 則概率不變.Binomial Distribution二項(xiàng)式分布Page 34陽山書屋cThe location, dispersion and shape of a binomial distribution are affected by the sample size, n, and defective rate, p.二項(xiàng)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(xiàng)式分布的參數(shù)分布參數(shù)Binomial Distribution二項(xiàng)式分布Page 35陽山書屋cNor

39、mal Approximation to the Binomial二項(xiàng)式分布的正態(tài)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,二項(xiàng)式分布是一個(gè)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

40、approaches the shape of a normal distribution:在一定的情況下(n 和p一定),二項(xiàng)式分布近似于一個(gè)正態(tài)分布的形狀.For p 0.5,np 5For p far from 0.5 (smaller or larger),np 10Binomial Distribution二項(xiàng)式分布Page 36陽山書屋cMean and Variance 均值和方差A(yù)lthough 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.盡管n 和 p 給定了一個(gè)特定的二項(xiàng)式分布,但分布的均值和方差經(jīng)常被用于實(shí)際的分布,象p-chart.The mean and variance of a binomial distribution二項(xiàng)式分布的均值和方差orBinomial Distribution二項(xiàng)式分布Page 37陽山書屋cImportantDiscrete Distributions重要的離散分布Binomial Distribution 二項(xiàng)式分布Poisson Distribution 泊松

42、分布Page 38陽山書屋cPoisson Distribution泊松分布This distribution have been found to be relevant for applications involving error rates, particle count, chemical concentration, etc,此分布被發(fā)現(xiàn)應(yīng)用于錯(cuò)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(gè)單位或空間中事件(或

43、缺陷率)的平均數(shù)量.And where is small.Page 39陽山書屋cSimeon D PoissonPage 40陽山書屋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)單位時(shí)間(或空間)的數(shù)量輸出獨(dú)立于另一個(gè)單位時(shí)間(或空間)的數(shù)量輸出.probability of an occurrence wit

44、hin a very short time interval (or space region) is proportional to the time interval (or space region)在非常短時(shí)間(或空間)內(nèi)發(fā)生的概率是單位時(shí)間(或單位空間)輸出數(shù)量的比率probability of more than 1 outcome occurring within a short time interval (or space region) is negligible極短時(shí)間(空間單位)內(nèi)1個(gè)數(shù)量輸出的概率可忽略不記the mean and variance for a Poi

45、sson Distribution are泊松分布的均值和方差是andPage 41陽山書屋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,離散和形狀都受均值影響Page 42陽山書屋cExample 2練習(xí)2.A certain process yields a defect rate of 4 dpmo. For a million opportunities inspected, determine t

46、he probability distribution.某一工序產(chǎn)生的缺陷率是4dpmo. 試計(jì)算其概率分布.Page 43陽山書屋cExample 2Calc Probability Distributions Poissona) Probability Mass Function b) Cumulative Distribution FunctionPage 44陽山書屋cSummary of Approximations近似總結(jié)Binomial p 5if p 5np 10 if |p| Poisson Normal Page 45陽山書屋cImportantContinuous Dis

47、tributionsNormal DistributionExponential DistributionPage 46陽山書屋cNormal Distribution正態(tài)分布Normal DistributionPage 47陽山書屋cThe most widely used model for the distribution of continuous random variables.連續(xù)性隨機(jī)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 velocity of m

48、olecules, 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.Normal Distribution正態(tài)分布Page 48陽山書屋cMany natural phenomena and man-made processes are observed to have normal distributions, or can be closely represented as normally d

49、istributed.我們觀測(cè)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(jī)器元件的長(zhǎng)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 variat

50、ions, raw material changes and contamination level changes溫度漂移,濕度變化,振動(dòng),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(jí)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(dú)立和近似可能相對(duì)于均值偏正或偏負(fù),則長(zhǎng)度近似于一個(gè)正態(tài)分布.Normal Distribution正態(tài)分布Page 49陽山書屋cCumulative Distribution Functio