Environmentally Conscious Design & Manufacturing

1. Environmentally Conscious Design & Manufacturing Class25: Probability and Statistics Prof. S. M. Pandit

2. Agenda • Random Variable • Mean and variance • Normal distribution • Sampling • Linear regression

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 x1, x2,…, xn and no other value. • Continuous random variable: the random variable X can take on a nondenumerably infinite number of values.

4. Continuous Random Variables The cumulative distribution function F(x):

5. Continuous Random Variables The probability density function f(x): The probability of occurrence of interval [a,b]:

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. 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. 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

9. Normal Distribution Probability density

10. Normal Distribution 99.73% 95.45% 68.27% -3 -2 -  + +2 +3

11. Standard Normal Distribution Let Cumulative probabilities

12. Standard Normal Distribution

13. Sampling Population and sample Population and sample Population N Population  Sample N Sample average Sample variance Sd is the sample standard deviation

14. Estimator of Sample Mean & Variance

15. Confidence Interval 90%, 95%, 99% probability limits on X are (1-) % confidence interval on the mean is where

16. Data Example: Grinding Wheel Profile

17. Linear Regression To express the dependence of one set of observations yt on another setxt under the assumption thatyt’s are independent or uncorrelated. model “best fit” where

18. Least Squares Estimates To minimize the sum of squares of the “ residuals” t’s Let NID-Normally Independently Distributed

19. Simple Linear Regression

20. Normal Distribution of yt Observation=Prediction+Error

21. Computations Examples Removing the mean yields

22. Computations Examples

23. Computations Examples Assuming that these estimated values are true values The % 95 probability limits for the observation yt are

24. Regression Equation with Observed Data

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.