lecture5 - http/ocw.mit.edu MIT OpenCourseWare 2.830J...

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MIT OpenCourseWare _________ http://ocw.mit.edu _ __ 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .
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1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #5 Probability Models, Parameter Estimation, and Sampling February 21, 2008
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2 M anufacturing The Normal Distribution p ( x ) = 1 σ 2 π e 1 2 x μ 2 z 0 0.1 0.2 0.3 0.4 -4 -3 -2 -1 0 1 2 3 4 z = x − μ “Standard normal” z =0 z =1
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3 M anufacturing Properties of the Normal pdf • Symmetric about mean • Only two parameters: μ and σ • Mean ( μ ) and Variance ( σ 2 ) have well known “estimators” (average and sample variance) p ( x ) = 1 σ 2 π e 1 2 x − μ 2
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4 M anufacturing Testing 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 3 σ bands • Confidence Intervals – Look at “kurtosis” • Measure of deviation from normal – Probability (or qq) plots (see Mont. 3-3.7 or MATLAB stats toolbox)
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5 M anufacturing Kurtosis: Deviation from Normal k = n ( n + 1) ( n 1)( n 2)( n 3) x i x s 4 3( n 1) 2 ( n 2)( n 3) For sampled data: k = 0 -norma l k > 0 - more “peaked” k < 0 - more “flat”
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6 M anufacturing Kurtosis for Some Common Distributions D: Laplace ( k = 3) L: logistic ( k = 1.2) N: normal ( k = 0) U: uniform ( k = -1.2) Source: Wikimedia Commons, http://commons.wikimedia.org
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7 M anufacturing •P lo t – normalized (mean centered and scaled to s ) vs. – theoretical position of unit normal distribution for ordered data • Normal distribution: data should fall along line Quantile-Quantile (qq) Plots Source: Wikimedia Commons, http://commons.wikimedia.org
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8 M anufacturing Guaranteeing “Normality” The Central Limit Theorem – If x 1 , x 2 ,x 3 . .. x N are N independent observations of a random variable with “moments” μ x and σ 2 x , – The distribution of the sum of all the samples will tend toward normal.
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9 M anufacturing Example: Uniformly Distributed Data 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 x 1 x 2 x 100 + + . . . Sum of 100 sets of 1000 points each 35 40 45 50 55 60 65 0 50 100 150 y = x i i = 1 100
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10 M anufacturing Sampling: 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.
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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lecture5 - http/ocw.mit.edu MIT OpenCourseWare 2.830J...

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