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Environmentally Conscious Design & Manufacturing Class 25: Probability and Statistics Prof. S. M. Pandit Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:1 Agenda • Random Variable • Mean and variance • Normal distribution • Sampling • Linear regression Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:2 Random Variables Random variables are numerical-valued quantities whose observed values are governed by the laws of probability. • • Discrete random variables: the random variable X can take on only one of several discrete values x1, x2,…, xn and no other value. Continuous random variable: the random variable X can take on a nondenumerably infinite number of values. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:3 Continuous Random Variables The cumulative distribution function F(x): F(x) P(X x) 0 F(x) 1 F(x1 ) F(x 2 ) if x1 x 2 ; lim F(x) 1 x lim F(x) 0 x Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:4 Continuous Random Variables The probability density function f(x): d f(x) F(x) dx The probability of occurrence of interval [a,b]: b P(a X b) f(x)dx F(b) - F(a) a Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:5 Moments of Random Variables The moments of the distribution mr r x f(x)dx The first -order moment is called the mean, the expected value or the expectation E[X]: E X xf(x)dx μ x 2 The second -order moment is called the variance σ x : σ x2 (x -μ x ) 2 f(x)dx E(X - μ x ) 2 0 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:6 Properties of Expectation E(cX)=cE(X), where c is a constant E(X+Y)=E(X)+E(Y) E(XY)=E(X)E(Y) if X & Y are independent Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:7 Theorems on Variance Var(cX)=c2Var(X) Var(X+Y)=Var(X)+Var(Y) (s2x+y=s2x+s2y) if X and Y are independent Var(X-Y)=Var(X)+Var(Y) (s2x-y=s2x+s2y) if X and Y are independent Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:8 Normal Distribution Probability density 1 f(x) e 2π (x μ x ) 2 2σ x 2 ,σ x 0, μ x b P(a X b) f(x)dx F(b) - F(a) a P(- X ) f(x)dx 1 - Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:9 Normal Distribution 99.73% 95.45% 68.27% -3 -2 - + +2 +3 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:10 Standard Normal Distribution Let Z X μx σx 1 μ z E(Z) (μ x μ x ) 0 σx Z ~ N(0,1) σ z2 E(Z μ z )2 E(Z 2 ) 1 Cumulative probabilities Φ(z) P( z ) Φ(1.65) 0.95053 Φ(1.96) 0.975 Φ(2.58) 0.99506 P( 1.65 Z 1.65) Φ(1.65) Φ( 1.65) Φ(1.65) (1 Φ( 1.65)) 90% Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:11 Standard Normal Distribution P( 1.65 Z 1.65) Φ(1.65) Φ( 1.65) Φ(1.65) (1 Φ( 1.65)) 90% P( 1.96 Z 1.96) Φ(1.96) Φ( 1.96) Φ(1.96) (1 Φ( 1.96)) 95% P( 2.58 Z 2.58) Φ(2.58) Φ( 2.58) Φ(2.58) (1 Φ( 2.58)) 99% Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:12 Sampling Population and sample Population N μ x E(X), σ x2 E(X μ x )2 X Sample average Sample N N 1 Xi N i 1 1 N x xi N i 1 Sample variance 1 n 2 S (x x ) i N 1 i 1 2 d Sd is the sample standard deviation Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:13 Estimator of Sample Mean & Variance E(X) E( 1 1 1 1 X ) E( X X ... XN ) i 1 2 N N N N 1 1 1 μ μ μ E(X 1 ) E(X 2 ) ... E(X N ) X X ... X μ X N N N N N N 1 1 1 X 1 X 2 ... X N ) N N N 1 1 1 NVar(X) Var(X) 2 Var(X 1 ) 2 Var(X 2 ) ... 2 Var(X N ) 2 N N N N N Var(X) Var( 1 σ E(X E(X)) σ X2 N 2 X 2 σx 1 σx N Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:14 Confidence Interval X μ x Zσ x mean Z * standard deviation 90%, 95%, 99% probability limits on X are μ x 1.65σ x , μx 1.96σ x , μ x 2.58σ x σ x2 X ~ N(μ x , ) N (1-) % confidence interval on the mean μ x is x zα/2 σ x / N where Φ(zα/2 ) 1 - α/2 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:15 Data Example: Grinding Wheel Profile Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:16 Linear Regression To express the dependence of one set of observations yt on another set xt under the assumption that yt’s are independent or uncorrelated. model y t β0 β1 x t εt , t 1,2,..., N N “best fit” βˆ 0 y βˆ1 x and βˆ1 (y t 1 t y)(x t x) N 2 ( x x ) t t 1 where 1 N 1 N x x t , y y t N i 1 N i 1 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:17 Least Squares Estimates To minimize the sum of squares of the “ residuals” t’s Let y t y y t , x t x xt y t β1 x t ε t , t 1,2,..., N, εt ~ NID(0,σ ε2 ) NID-Normally Independently Distributed N β̂1 y x t 1 N t xt 2 t N 1 residual sum of squares 2 ˆ ˆ σ (y t β1 x t ) N t 1 Number of residuals 2 ε t 1 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:18 Simple Linear Regression Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:19 Normal Distribution of yt Observation=Prediction+Error y t β1 x t ε t yˆ t ε t Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:20 Computations Examples t 1 2 3 4 5 N 5 x t 5 6 3 2 5 y t 7 6 5 4 6 1 1 x t 5 (5 6 3 2 5) 4.2 N t 1 1 y y t (7 6 5 4 6) 5.6 N t 5 x Removing the mean yields t 1 2 3 4 5 x t 0.8 1.8 - 1.2 - 2.2 0.8 y t 1.4 0.4 - 0.6 - 1.6 0.4 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:21 Computations Examples N βˆ1 y x t 1 N t xt 2 t 6.4 0.59 10.8 t 1 5 1 σˆ (y t 0.59x t ) 2 5 t 1 1 [(0.928 2 ( 0.662) 2 (0.108) 2 ( 0.302) 2 ( 0.072) 2 0.281 5 2 ε Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:22 Computations Examples Assuming that these estimated values are true values y t β1 x t ε t 0.59x t ε t ε t ~ NID(0,0.281) ε t y t 0.59x t yˆ t β1 x t 0.59x t The % 95 probability limits for the observation yt are yˆ t 1.96σε β1 xt 1.96σε 0.59x t 1.96 0.281 0.59x t 1.04 Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:23 Regression Equation with Observed Data Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:24 Homework #8 The problems 1 through 2 are out of the textbook “Industrial Ecology” 1. Problem 14.2 (Answer: Φ 37.8GJ/t, Ω 0.77, Ψ 0.23 ) 2. Problem 14.3 (Answer: Φ 43.8075GJ/ t, Ω 0.9625, Ψ 0.23 ) 3. List some of the new environmentally friendly energy technologies. 4. Discuss and illustrate the contrast between the traditional and loss function based approaches to characterize quality. 5. Discuss and illustrate the shortcomings of the loss function approach and how they can be overcome by the satisfaction metric that includes benefits. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:25