The Normal DistributionProperties of the Normal pdfTesting the Model: e.g. Is the Process “Normal” ? Kurtosis: Deviation from NormalKurtosis for Some Common DistributionsQuantile-Quantile (qq) PlotsGuaranteeing “Normality” The Central Limit TheoremExample: Uniformly Distributed DataSampling: Using Measurements (Data) to Model the Random ProcessSample StatisticsSample Mean Uncertainty Manufacturing as Random Processes Formal Use of Statistical ModelsDiscrete Distribution: BernoulliDiscrete Distribution: BinomialBinomial DistributionDiscrete Distribution: PoissonPoisson DistributionsBack to Continuous DistributionsContinuous Distribution: UniformStandard Questions For a Known cdf or pdfContinuous Distribution: Normal or GaussianContinuous Distribution: Unit NormalUsing the Unit Normal pdf and cdfUse of the pdf: Location of DataLocation of DataLocation of DataStatisticsSampling to Determine Parameters of the Parent Probability DistributionMoments of the Population vs. Sample StatisticsSampling and EstimationPopulation vs. Sampling DistributionSampling and Estimation, cont.Sampling and Estimation – An ExampleEstimation and Confidence IntervalsConfidence Intervals: Variance KnownExample, Cont’dSummaryMIT OpenCourseWare _________http://ocw.mit.edu___ 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)Spring 2008For information about citing these materials or our Terms of Use, visit: ________________http://ocw.mit.edu/terms.1ManufacturingControl of Manufacturing ProcessesSubject 2.830/6.780/ESD.63Spring 2008Lecture #5Probability Models, Parameter Estimation, and SamplingFebruary 21, 20082ManufacturingThe Normal Distributionp(x) =1σ2πe−12x−μσ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2z00.10.20.30.4-4 -3 -2 -1 0 1 2 3 4z =x −μσ“Standard normal”μz=0σz=13ManufacturingProperties of the Normal pdf• Symmetric about mean• Only two parameters:μ and σ• Mean (μ) and Variance ( σ2) have well known “estimators” (average and sample variance)p(x) =1σ2πe−12x−μσ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 24ManufacturingTesting the Model: e.g. Is the Process “Normal” ?• Is the underlying distribution really normal?– Look at histogram– Look at curve fit to histogram– Look at % of data in 1, 2 and 3σ bands• Confidence Intervals– Look at “kurtosis”• Measure of deviation from normal– Probability (or qq) plots (see Mont. 3-3.7 or MATLAB stats toolbox)5ManufacturingKurtosis: Deviation from Normalk =n(n +1)(n −1)(n − 2)(n − 3)xi− xs⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∑4⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −3(n −1)2(n − 2)(n − 3)For sampled data:k = 0 -normalk > 0 - more “peaked”k < 0 - more “flat”6ManufacturingKurtosis for Some Common DistributionsD: Laplace (k = 3)L: logistic (k = 1.2)N: normal (k = 0)U: uniform (k = -1.2)Source: Wikimedia Commons, http://commons.wikimedia.org7Manufacturing•Plot– normalized (mean centered and scaled to s)vs. – theoretical position of unit normal distribution for ordered data• Normal distribution: data should fall along lineQuantile-Quantile (qq) PlotsSource: Wikimedia Commons, http://commons.wikimedia.org8ManufacturingGuaranteeing “Normality”The Central Limit Theorem– If x1, x2,x3 ...xN… are N independent observations of a random variable with “moments”μxand σ2x,– The distribution of the sum of all the samples will tend toward normal.9ManufacturingExample: Uniformly Distributed Data0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10102030405060700 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10102030405060700 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1010203040506070x1x2x100++ . . . Sum of 100 sets of 1000 points each35 40 45 50 55 60 65050100150y = xii=1100∑10ManufacturingSampling: Using Measurements (Data) to Model the Random Process• In general p(x) is unknown• Data can suggest form of p(x)– e.g.. uniform, normal, weibull, etc.• Data can be used to estimate parameters of distributions–e.g. μand σfor normal distribution: p(x) = N(μ,σ2)• How to estimate– Sample Statistics• Uncertainty in estimates– Sample Statistic pdf’sp(x) =1σ2πe−12x−μσ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 211ManufacturingSample StatisticsAverage or sample meanSample varianceSample standard deviation12ManufacturingSample Mean Uncertainty • If all xicome from a distribution with μxand σ2x, and we divide the sum by n:x =1nxii=1n∑μx=μxσx2=1nσx2 or σx=1nσxx=c1x1+c2x2+ c3x3+K cnxn ci=1nThen:and13ManufacturingManufacturing as Random Processes • All physical processes have a degree of natural randomness• We can model this behavior using probability distribution functions• We can calibrate and evaluate the quality of this model from measurement data14ManufacturingFormal Use of Statistical Models• Discrete Variable Distributions and Uses– Attribute Modeling• Sampling: Key distributions arising in sampling• Chi-square, t, and F distributions• Estimation: – Reasoning about the population based on a sample• Some basic confidence intervals• Estimate of mean with variance known• Estimate of mean with variance not known• Estimate of variance• Hypothesis tests15ManufacturingDiscrete Distribution: BernoulliBernoulli trial: an experiment with two outcomesProbability density function (pdf):xf(x)p1 -p01¼¾16ManufacturingDiscrete Distribution: BinomialRepeated random Bernoulli trials• n is the number of trials• p is the probability of “success” on any one trial• x is the number of successes in n trials17ManufacturingBinomial DistributionBinomial Distribution00.050.10.150.20.251357911131517192123252729Number of "Successes"00.20.40.60.811.21357911131517192123252729Series1e.g. expected number of “rejects”18ManufacturingDiscrete Distribution: Poisson• Poisson is a good approximation to Binomial when n is large and p is small (< 0.1)Example applications:# misprints on page(s) of a book# defects on a waferMean:Variance:19ManufacturingPoisson Distribution00.010.020.030.040.050.060.070.080.090.11357911131517192123252729313335373941Events per unitcPoisson Distribution00.020.040.060.080.10.120.140.160.180.21 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Events per unitcPoisson Distribution00.010.020.030.040.050.060.070.081357911131517192123252729313335373941Events per unitλ=5λ=20λ=30Poisson Distributionse.g. defects/device20ManufacturingBack to Continuous Distributions• Uniform Distribution• Normal Distribution– Unit (Standard) Normal