1、Probability Distributions概率分布,MPT Confidential,Page 2,Learning Objectives學習目的,What is a Probability Distribution? 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 Discrete Distributions離散分布 Binomial Distribution 二項式分布 Poisson Distrib

2、ution 泊松分布. Hypergeometric 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 指數分布 Logarithmic normal distribution 對數正態(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 Distribu

3、tion c2 分布 F Distribution F 分布,MPT Confidential,Page 3,As we progress from description of data towards inference of data, an important concept is the idea of a probability distribution. 當我們從描述性數據進步到推論性數據時,一個重要的內容就是概率分布的概念. To appreciate the notion of a probability distribution, we need to review var

4、ious fundamental concepts related to it: 為了解概率分布的概念, 我們需要復習各種基本相關概念: Experiment, Sample Space, Event 實驗,樣本空間,事件 Random Variable 隨機變量.,What is a Probability Distribution?什么是概率分布?,What do we mean by inference of data?,MPT Confidential,Page 4,Experiment實驗 An experiment is any activity that generates a

5、set of data, which may be numerical or not numerical. 實驗是產生一系列數據的行為,數據有可能是數字的或非數字的.,1, 2, ., 6,(a),Throwing a dice 擲骰子,Experiment generates numerical / discrete data,(b),Inspecting for stain marks檢查污點印記,Experiment generates attribute data,(c),Measuring shaft 測量 軸徑,Experiment generates continuous dat

6、a,What is a Probability Distribution?什么是概率分布?,實驗產生數字/離散數據,實驗產生計數性數據,實驗產生連續(xù)性數據,MPT Confidential,Page 5,Random Experiment 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 cer

7、tainty. 如果我們擲子一次由一次,或從相同工序生產許多軸,結果會是不同的.不能完全提前預測. An experiment which can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. 一個實驗導致不同的結果,即使它是每次以相同方式,這叫做隨機實驗,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 6,Sample

8、Space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. 一個結果,或一套結果,從一個隨機實驗出來的稱為事件,也就是樣本空間的子集,What is a Probability D

9、istribution?什么是概率分布?,MPT Confidential,Page 7,Event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 結果是奇數 Event事件 2: the outcome is a number 4 大于4的結果 Example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

10、徑大于平均值 Event 事件2: the outcome is a part failing specs.未通過規(guī)格的結果., E2 = x USL, E2 = 5, 6, E1 = 1, 3, 5, E1= x m,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 8,Random Variable隨機變量 From a same experiment, different events can be derived depending on which aspects of the experiment w

11、e 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生不同的結果 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方面, 通過一個大寫的字母表示.舉例說明: Let X be t

12、he event “the number of a dice is odd”. 用X代表事件”骰子的數字是奇數” 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”,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 9,Random Variable隨機變量 We have defined a function that assigns a real number to an experimental outcome within t

13、he sample space of the random experiment. 我們定義了一個函數,其代表了一個在隨機實驗的樣本空間的一個真實實驗數字 This function (X or W in our examples) is called a random variable because: 函數(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)生結果是明顯不定的,是同另一個結果

14、相異的. Each outcome has an equal chance of being selected. 每一個結果有相同被選擇的機會.,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 10,Probability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%)

15、. 為量化一個隨機變量的指定結果發(fā)生的可能性,我們指定一個數字介于0和1之間(或0100%) This numerical value is called the probability of the outcome. 這個數字稱為結果的概率 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 sa

16、me 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后發(fā)生的分數(或比例)次數 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,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 11,Probability Distribution概率分布 When we are a

17、ble 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 is called a probability distribution of X. 當我們能夠表明一個隨機變量的某一個可能結果的概率,則整個可能結果的概率的描述稱為X的概率分布 A probability distribution is typically presented

18、 as a curve or plot that has: 一個概率分布被代表為一個曲線或點應有: All the possible outcomes of X on the horizontal axis X的所有的可能結果在水平軸線上 The probability of each outcome on the vertical axis 每一個結果的概率在縱軸上,What is a Probability Distribution?什么是概率分布?,MPT Confidential,Page 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

19、等 隨機事件 事件間的關系 互斥 事件的運算： 對立、并、交、差,關于概率,MPT Confidential,Page 13,Normal Distribution,Exponential Distribution,Uniform Distribution,Binomial Distribution,Discrete 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 Dis

20、tribution?什么是概率分布?,MPT Confidential,Page 14,Empirical Distributions經驗分布,Created from actual observations. Usually represented as histograms. 根據實際觀測得來, 通常用直方圖代表 Empirical distributions, like theoretical distributions, apply to both discrete and continuous distributions. 經驗分布,象理論上的分布,適用于離散和連續(xù)分布.,MPT C

21、onfidential,Page 15,Three 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 - 定義中心趨勢的數據 Spread分布(或離散,或刻度)- defines dispersion of data (or Dispersion, or Scale) 定義數據的離散,Properties of Distributions分布的描述,Exponential D

22、istribution,統(tǒng)一分布,指數分布,MPT Confidential,Page 16,Shape形狀 Describes how the probabilities of all the possible outcomes are distributed. 描述所有可能結果可能性的分布 Can be described mathematically with an equation called a probability function, e.g: 可以用一個概率函數數字表示,舉例說明,Probability function 概率函數,Lowercase letter repre

23、sents 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),Properties of Distributions分布的描述,MPT Confidential,Page 17,0,0,f(t),1a,2a,3a,b = 4,2,1,0.5,Probability Functions概率函數 For a discrete distribution,對于一個離散分布 f(x) called is the probability f(x) 稱為概率集中: mass function (pmf), e

24、.g.:函數,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.:函數舉例說明,Properties of Distributions分布的描述,MPT Confidential,Page 18,The total probability for any distribution sums to 1. 任何分布的全部概率總和為1 In a discrete distribution, probability is r

25、epresented as height of the bar. 在一個離散分布,概率用柱狀表示 In a continuous distribution, probability is represented as 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,Properties of Distributions分布的描述,MPT Confidential,Page 19,Probability of An Exact Value Under PDF is Zero! PDF下一個準確值的概率是

26、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 variable X, how do we interpret the

27、 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, 我們如何解釋12.57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 obtained by integrating the area und

28、er 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) = 0,for a continuous random variable,Properties of Distributions分布的描述,MPT Confidential,Page 20,Exponential Distribution,Area of a line is zero! f(9.5) = P(X = 9.5) = 0,To get probability of 20.0, integrate area between

29、 19.995 and 20.005, i.e. P(19.995 X 20.005),Area denotes probability of getting a value between 40.0 and 50.0.,Note: f(x) is used to calculate an area that represents probability 注意:f(x) 用于計算一個代表概率的面積,Properties of Distributions分布的描述,MPT Confidential,Page 21,Instead of a probability distribution fun

30、ction, it is often useful to describe, for a specific value x of a random variable, the total probability of all possible values occurring, up to result of a unit does not influence outcome of next unit 試驗是獨立的，一個單位的結果不影 響下一個結果的輸出。,Each trial results in only two possible outcomes. 每一次試驗只有兩種可能的結果。,A b

31、inomial experiment! 一個二項式試驗,MPT Confidential,Page 29,Probability Mass Function概率集中函數 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? 如果每一個二項式實驗（隨機取n 個產品進行通過/拒收檢查）被重復很多次，我們是否可以每次看到相同的X不合格品 Th

32、e 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,Probability of getting x defective units (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個樣本量（n次）,Given that the overall defective rate is p (proba

33、bility of success is p) 給出整個不合格品率p (成功的概率是P）,Binomial Distribution二項式分布,MPT Confidential,Page 30,Applications應用 The binomial distribution is extensively used to model results of experiments that generate binary outcomes, e.g. pass/fail, go/nogo, accept/reject, etc. 二項式分布廣泛應用于結果只輸出兩種的實驗.舉例來說,通過/不通過,去

34、/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,常用于缺陷品計數的數據,舉例來說 1. Acceptance Sampling 接受樣本 2. p-chart P-Chart,Binomial Distribution二項式分布,MPT Confidential,Page 31,Example 1例1 If a process historically gives 10% reject rate (p = 0.10), 如果

35、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.,Binomial Distribution二項式分布,MPT Confidential,Page 32,Example 1 (contd)例1繼續(xù) These probabilities can be obtained from Minitab: 這些概率可通過Minitab獲得: Cal

36、c Probability Distributions Binomial,P(x),n = 20,p = 0.1,包含X個缺陷品的指定列,存儲結果的指定列,Binomial Distribution二項式分布,MPT Confidential,Page 33,Example 1 (contd),From Excel:,From Minitab:,What is the probability of getting 2 defectives or less?,Binomial Distribution二項式分布,MPT Confidential,Page 34,Example 1 (contd)

37、例1(繼續(xù)) For the 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量 X If we divide each x value by constant sample size, n, and re-express the x-axis as a proportion defective p-axis, the probabilities do 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.,Bin

38、omial Distribution二項式分布,MPT Confidential,Page 35,The location, 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,分布參數,Binomial Distribution二項式分布,MPT Confidential,Page

39、36,Normal Approximation to the Binomial 二項式分布的正態(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 appr

40、oximately 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,np 5,For p far from 0.5 (smaller or larger),np 10,Binomial Distribution二項式分布,MPT Confidential,Page 37,Mean and Variance 均值和方差 Although n and p pin down a specific binomial distribution, often the me

41、an 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常被用于實際的分布,象p-chart. The mean and variance of a binomial distribution 二項式分布的均值和方差,or,Binomial Distribution二項式分布,Important Discrete Distributions 重要的離散分布,Binomial Distribution 二項

42、式分布 Poisson Distribution 泊松分布,MPT Confidential,Page 39,Poisson 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)應用于錯誤率,灰塵數,化學比,等等. where is the mean number of events (or defect rate) within a

43、given unit of time or space. 是給定的一個單位或空間中事件(或缺陷率)的平均數量.,And where is small.,MPT Confidential,Page 40,Simeon D Poisson,MPT Confidential,Page 41,Poisson 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 spac

44、e region) 單位時間(或空間)的數量輸出獨立于另一個單位時間(或空間)的數量輸出. probability of an occurrence within a very short time interval (or space region) is proportional to the time interval (or space region) 在非常短時間(或空間)內發(fā)生的概率是單位時間(或單位空間)輸出數量的比率 probability of more than 1 outcome occurring within a short time interval (or spa

45、ce region) is negligible 極短時間(空間單位)內1個數量輸出的概率可忽略不記 the mean and variance for a Poisson Distribution are 泊松分布的均值和方差是,MPT Confidential,Page 42,Poisson 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,MPT Confidential,Page 43,Exa

46、mple 2練習2.,A certain process yields a defect rate of 4 dpmo. For a million opportunities inspected, determine the probability distribution. 某一工序產生的缺陷率是4dpmo. 試計算其概率分布.,MPT Confidential,Page 44,Example 2,Calc Probability Distributions Poisson a) Probability Mass Function b) Cumulative Distribution Function,MPT Confidential,Page 45,Summary of Approximations近似總結,Binomial,p 0.1,the smaller the p 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,MPT Confidential,Page 49,Many natural phe