《西格玛应用》PPT课件.ppt

1,Probability Distributions,What we have learned thus far ,Tools Learned Descriptive Statistics Capability Study Process Mapping Pareto Chart Fishbone Diagram FMEA,Descriptive Statistics,comprises those methods concerned with collecting and describing a set of data so as to yield meaning information,We are crossing the bridge now ,Inferential Statistics,Descriptive Statistics,Statistics Overview,What will be taught.,Tools to be taught Parameter Estimation Hypothesis Testing ANOVA,Inferential Statistics,comprises those methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data,Learning Objectives,What is a Probability Distribution? Experiment, Sample Space, Event Random Variable, Probability Functions (pmf, pdf, cdf) Discrete Distributions Binomial Distribution Poisson Distribution Continuous Distributions Normal Distribution Exponential Distribution Sampling Distributions Z Distribution t Distribution c2 Distribution F Distribution,As we progress from description of data towards inference of data, an important concept is the idea of a probability distribution. To appreciate the notion of a probability distribution, we need to review various fundamental concepts related to it: Experiment, Sample Space, Event Random Variable,What is a Probability Distribution?,What do we mean by inference of data?,Experiment An experiment is any activity that generates a set of data, which may be numerical or not numerical.,What is a Probability Distribution?,1, 2, , 6,(a),Throwing a dice,Experiment generates numerical / discrete data,(b),Inspecting for stain marks,Experiment generates attribute data,(c),Measuring shaft ,Experiment generates continuous data,Random Experiment 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. An experiment which can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment.,What is a Probability Distribution?,Sample Space The collection of all possible outcomes of an experiment is called its sample space. 1, 2, , 6 - Tossing of a dice Pass, Fail - Inspecting for stain marks All possible values - Measuring shaft between 0 and 10mm) Event 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.,What is a Probability Distribution?,Event Example 1: Some events from tossing of a dice. Event 1: the outcome is an odd number Event 2: the outcome is a number 4 Example 2: Some events from measuring shaft : Event 1: the outcome is a diameter mean Event 2: the outcome is a part failing specs.,What is a Probability Distribution?, E2 = x USL, E2 = 5, 6, E1 = 1, 3, 5, E1= x m,Random Variable From a same experiment, different events can be derived depending on which aspects of the experiment we consider important. 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.: Let X be the event “the number of a dice is odd”. Let W be the event “the shaft is within specs.”.,What is a Probability Distribution?,Random Variable We have defined a function that assigns a real number to an experimental outcome within the sample space of the random experiment. This function (X or W in our examples) is called a random variable because: The outcomes of the same event are clearly uncertain and are variable from one outcome to another. Each outcome has an equal chance of being selected.,What is a Probability Distribution?,Probability 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%). This numerical value is called the probability of the outcome. 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. This method is the relative frequency approach or frequentist approach to interpreting probability.,What is a Probability Distribution?,Probability Distribution 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 is called a probability distribution of X. A probability distribution is typically presented as a curve or plot that has: All the possible outcomes of X on the horizontal axis The probability of each outcome on the vertical axis,What is a Probability Distribution?,Normal Distribution,Exponential Distribution,Uniform Distribution,Binomial Distribution,Discrete Probability Distributions (Theoretical),What is a Probability Distribution?,Continuous Probability Distributions (Theoretical),Empirical Distributions,Created from actual observations. Usually represented as histograms. Empirical distributions, like theoretical distributions, apply to both discrete and continuous distributions.,Three common important characteristics: Shape - defines nature of distribution Center - defines central tendency of data Spread - defines dispersion of data (or Dispersion, or Scale),Properties of Distributions,Exponential Distribution,Shape Describes how the probabilities of all the possible outcomes are distributed. Can be described mathematically with an equation called a probability function, e.g:,Properties of Distributions,Probability function,Lowercase letter represents a specific value of random variable X,f(x) means P(X = x),Probability Functions For a discrete distribution, f(x) called is the probability mass function (pmf), e.g.: For a continuous distribution, f(x) is called the probability density function (pdf), e.g.:,Properties of Distributions,Properties of Distributions,The total probability for any distribution sums to 1. In a discrete distribution, probability is represented as height of the bar. In a continuous distribution, probability is represented as area under the curve (pdf), between two points.,Properties of Distributions,Probability of An Exact Value Under PDF is Zero! 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: 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? It is interpreted as the probability of X assuming a value within a small interval around 12.57, i.e. 12.565, 12.575. This is obtained by integrating the area under the pdf between 12.565 and 12.575.,P(X = x) = 0,for a continuous random variable,Properties of Distributions,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 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,Instead of a probability distribution function, 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 & including x, i.e. P(X x). A equation or function that links a specific x value to the cumulated probabilities of all possible values up to and including x is called a cumulative distribution function (cdf), denoted as F(x). F(x) = P(X x),Cumulative Distribution Function,Compare against: f(x) = P(X = x),Cumulative Distribution Function,Normal Distribution,Discrete Distribution,Cumulative Distribution Function,Common Probability Distributions,Discrete Distributions Uniform Binomial Geometric Hypergeometric Poisson Continuous Distributions Uniform Normal Exponential Weibull Erlang, Gamma Lognormal,Theoretically derived distributions using certain random experiments that frequently arise in applications. Used to model outcomes of physical systems that behave similarly to random experiments used to derive the distributions.,28,Important Discrete Distributions,Binomial Distribution Poisson Distribution,29,Binomial Distribution,Binomial Experiment Assuming we have a process that is historically known to produce p reject rate. p can be used as the probability of finding a failed unit each time we draw a part from the process for inspection. Lets pull a sample of n parts randomly from a large population ( 10n) for inspection. Each part is classified as accept or reject.,Binomial Distribution,Reject rate = p,Sample size (n),Binomial Experiment Assuming we have a process that is historically known to produce p reject rate. p can be used as the probability of finding a failed unit each time we draw a part from the process for inspection. Lets pull a sample of n parts randomly from a large population ( 10n) for inspection. Each part is classified as accept or reject.,Binomial Distribution,For each trial (drawing a unit), the probability of success is constant.,Trials are independent; result of a unit does not influence outcome of next unit,Each trial results in only two possible outcomes.,A binomial experiment!,Probability Mass Function 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? The pmf that describes how the x defective units (called successes) are distributed is given as:,Binomial Distribution,Probability of getting x defective units (x successes),Using a sample size of n units (n trials),Given that the overall defective rate is p (probability of success is p),Applications The binomial distribution is extensively used to model results of experiments that generate binary outcomes, e.g. pass/fail, go/nogo, accept/reject, etc. In industrial practice, it is used for data generated from counting of defectives, e.g.: 1. Acceptance Sampling 2. p-chart,Binomial Distribution,Example 1 If a process historically gives 10% reject rate (p = 0.10), what is the chance of finding 0, 1, 2 or 3 defectives within a sample of 20 units (n = 20)? 1.,Binomial Distribution,Example 1 (contd) 2. These probabilities can be obtained from Minitab: Calc Probability Distributions Binomial,Binomial Distribution,P(x),n = 20,p = 0.1,Specify column containing x defectives,Specify column to store results,Example 1 (contd) 3. Create its pmf from Minitab and read off the answers:,Binomial Distribution,Specify column of possible outcomes, x: 0 to 20 defectives. Compute and store results, P(x), as shown previously: Calc Probability Distributions Binomial Create a chart of the pmf: Graph Chart,Display data labels,Select range of x to plot,Specify axis titles,Example 1 (contd),Binomial Distribution,From Excel:,From Minitab:,What is the probability of getting 2 defectives or less?,Example 1 (contd) For the 2 previous charts, the x-axis denotes the number of defective units, 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.,Binomial Distribution,The location, dispersion and shape of a binomial distribution are affected by the sample size, n, and defective rate, p.,Parameters of Binomial Distribution,Binomial Distribution,Binomial Distribution,Normal Approximation to the Binomial 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. Under certain conditions (combinations of n and p), the binomial distribution approximately approaches the shape of a normal distribution:,For p 0.5, np 5,For p far from 0.5 (smaller or larger), np 10,Binomial Distribution,Mean and Variance Although 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. The mean and variance of a binomial distribution:,42,Poisson Distribution,Applications The Poisson distribution is a useful model for any random phenomenon that occurs on a per unit basis: Per unit area Per unit volume Per unit time, etc. A typical application is as a model of number of defects in a unit of product, e.g.: Number of cracks per 10m roll inspection in production of continuous rolls of sheetmetal. Number of particles per cm2 of part,Poisson Distribution,Defect rate = l,Inspection units (n),# defects per unit,Poisson Process The Poisson distribution is derived based on a random experiment called a Poisson process. Lets look at the inspection of 10m roll of sheetmetal again: Each 10m roll represents a subinterval of a continuous roll (interval) of real numbers. In the entire production of the continuous roll, defects occur randomly and results of an inspected unit does not influence outcome of the next unit.,Poisson Distribution,Conceptually, the sheetmetal can be partitioned into sub- intervals until each is small enough that: There is either 1 defect or none within the subinterval, i.e. probability of more than 1 defect is zero. Probability of 1 count in any subinterval is the same. Probability of 1 count increases proportionately as subinterval size increases.,Outcome in each.subinterval is independent of other subintervals,Probability Mass Function If each Poisson process (pulling an inspection unit randomly to count defects) is repeated over time, do we see the same x defects per unit all the time? The pmf that describes how the x defects (called counts) per unit are distributed is given as:,Poisson Distribution,Probability of getting x defects per inspection unit (x counts),Given that the overall defects per unit is l (defect rate is l),Example 2 If a process is historically known to give 4.0 defects per unit (l = 4), what is the chance of finding 0, 1, 2 or 3 defects per unit? 1.,Poisson Distribution,Poisson Distribution,Example 2 (contd) 2. These probabilities can be obtained from Minitab: Calc Probability Distributions Poisson,l = 4.0,P(x),Specify column containing x defects,Specify column to store results,Example 2 (contd) 3. Create its pmf from Minitab and read off the answers:,Poisson Distribution,Specify column of possible outcomes, x: 0 to 20 defects. Compute and store results, P(x), as shown previously: Calc Probability Distributions Binomial Create a chart of the pmf: Graph Chart,Display data labels,Select range of x to plot,Specify axis titles,Example 2 (contd),Poisson Distribution,From Excel:,From Minitab:,What is the probability of getting 2 defects or less?,The location, dispersion and shape of a Poisson distribution are affected by the mean, l.,Parameter of the Poisson Distribution,Poisson Distribution,Poisson Distribution,Normal Approximation to the Poisson The Poisson distributions are generally skewed to the right. For l 15, the Poisson distribution approximately approaches the normal distribution. Poisson Approximation to the Binomial The binomial distribution can be shown to approach the Poisson distribution in its limiting conditions, i.e.: when p is very small (approaching zero) n is large (approaching infinity) This allows the Poisson pmf to be used (easier) when a binomial experiment assumes above conditions,np = l = constant,Summary of Approximations,Binomial,p 0.1,The smaller the p & the larger the n the better, 15,The larger the better,np 10 p ,Poisson,Normal,Poisson Distribution,Mean and Variance Although l pins down a specific Poisson distribution, often the mean and variance of the distribution are used in practical applications such as the c-chart. The mean and variance of a Poisson distribution:,What happens to the variability as the mean of the Poisson distribution increases?,Exercises,A process yields a defective rate of 10%. For a sampling plan of 10 units, determine the probability distribution (pmf and cdf). A certain process yields a defect rate of 2.8 dpmo. For a million opportunities inspected, determine the probability distribution (pmf and cdf).,55,Important Continuous Distributions,Normal Distribution Exponential Distribution Weibull Distribution,56,Normal Distribution,Normal Distribution,The most widely used model for the distribution of continuous random variables. Arises 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.,Normal Distribution,Many natural phenomena and man-made processes are observed to have normal distributions, or can be closely represented as normally distributed. For example, the length of a machined part is observed to vary about its mean due to: temperature drift, humidity change, vibrations, cutting angle variations, cutting tool wear, bearing wear, rotational speed variations, fixturing variations, raw material changes and contamination level changes 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.,Normal Distribution,Cumulative Distribution Function,Normal Distribution,First introduced by French mathematician Abraham DeMoivre in 1733. Made famous in 1809 by German mathematician K.F. Gauss when he also developed a normal distribution independently and used it in his study of astronomy. As a result, it is also known as the Gaussian distribution.,Normal Distribution - Historical Notes,Karl Friedrich Gauss,During mid to late nineteenth century, many statisticians believed that it was “normal” for most well-behaved data to follow this curve, hence the “normal distribution”.,A normal distribution can be completely described by knowing only the: Mean (m) Variance (s2),Some Properties of the Normal Distribution,What is the difference between the 3