1. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 25: Probability and Statistics Prof. S. M. Pandit

2. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:2 Agenda Random Variable Mean and variance Normal distribution Sampling Linear regression

3. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:3 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 x 1, x 2,…, x n and no other value. Continuous random variable: the random variable X can take on a nondenumerably infinite number of values.

4. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:4 Continuous Random Variables The cumulative distribution function F(x):

5. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:5 Continuous Random Variables The probability density function f(x): The probability of occurrence of interval [a,b]:

6. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:6 Moments of Random Variables The moments of the distribution The first -order moment is called the mean, the expected value or the expectation E[X]: The second -order moment is called the variance :

7. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:7 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

8. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:8 Theorems on Variance Var(cX)=c 2 Var(X) Var(X+Y)=Var(X)+Var(Y) (s 2 x+y =s 2 x +s 2 y ) if X and Y are independent Var(X-Y)=Var(X)+Var(Y) (s 2 x-y =s 2 x +s 2 y ) if X and Y are independent

9. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:9 Normal Distribution Probability density

10. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:10 Normal Distribution  -3  68.27% 95.45% 99.73%  -2 --++ +2  +3 

11. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:11 Standard Normal Distribution Cumulative probabilities Let

12. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:12 Standard Normal Distribution

13. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:13 Population and sample Population N Population and sample Population  Sample N Sample average Sample variance S d is the sample standard deviation Sampling

14. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:14 Estimator of Sample Mean & Variance

15. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:15 Confidence Interval 90%, 95%, 99% probability limits on X are (1-  ) % confidence interval on the mean is where

16. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:16 Data Example: Grinding Wheel Profile

17. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:17 Linear Regression To express the dependence of one set of observations y t on another set x t under the assumption that y t ’s are independent or uncorrelated. where model “best fit”

18. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:18 Least Squares Estimates NID-Normally Independently Distributed To minimize the sum of squares of the “ residuals”  t ’s Let

19. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:19 Simple Linear Regression

20. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:20 Normal Distribution of y t Observation=Prediction+Error

21. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:21 Computations Examples Removing the mean yields

22. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:22 Computations Examples

23. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:23 Computations Examples Assuming that these estimated values are true values The % 95 probability limits for the observation y t are

24. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:24 Regression Equation with Observed Data

25. Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:25 Homework #8 The problems 1 through 2 are out of the textbook “Industrial Ecology” 1. Problem 14.2 (Answer: ) 2. Problem 14.3 (Answer: ) 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.