MATH408: Probability & Statistics Summer 1999 WEEK 5

1. MATH408: Probability & StatisticsSummer 1999WEEK 5 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu Homepage: www.kettering.edu/~schakrav

2. Joint PDF • So far we saw one random variable at a time. However, in practice, we often see situations where more than one variable at a time need to be studied. • For example, tensile strength (X) and diameter(Y) of a beam are of interest. • Diameter (X) and thickness(Y) of an injection-molded disk are of interest.

3. Joint PDF (Cont’d)X and Y are continuous • f(x,y) dx dy = P( x < X < x+dx, y < Y < y+dy) is the probability that the random variables X will take values in (x, x+dx) and Y will take values in (y,y+dy). • f(x,y)  0 for all x and y and

7. Measures of Joint PDF

13. Independence We say that two random variables X and Y are independent if and only if P(XA, YB) = P(XA)P(YB) for all A and B.

14. EXAMPLES

15. Groundwork for Inferential Statistics • Recall that, our primary concern is to make inference about the population under study. • Since we cannot study the entire population we rely on a subset of the population, called sample, to make inference. • We saw how to take samples. • Having taken the sample, how do we make inference on the population?

16. Basic Concepts

17. Figure 3-36(a) Probability density function of a pull-off force measurement in Example 3-33.

18. Figure 3-36 (b) Probability density function of the average of 8 pull-off force measurements in Example 3-33.

19. Figure 3-36 (c) Probability density Probability density function function of the sample variance of 8 pull-off force measurements in Example 3-33.

20. An important result

21. Examples

22. Central Limit Theorem • One of the most celebrated results in Probability and Statistics • History of CLT is fascinating and should read “The Life and Times of the Central Limit Theorem” by William J. Adams • Has found applications in many areas of science and engineering.

23. CLT (cont’d) • A great many random phenomena that arise in physical situations result from the combined actions of many individual ones. • Shot noise from electrons; holes in a vacuum tube or transistor; atmospheric noise, turbulence in a medium, thermal agitation of electrons in a conductor, ocean waves, fluctuations in stock market, etc.

24. CLT (cont’d) • Historically, the CLT was born out of the investigations of the theory of errors involved in measurements, mainly in astronomy. • Abraham de Moivre (1667-1754) obtained the first version. • Gauss, in the context of fitting curves, developed the method of Least Squares, which lead to normal distribution.

27. Examples

28. HOMEWORK PROBLEMS Sections 3.11 through 3.12 109,111, 114-116-119, 121-123, 129-130

34. Examples

35. Tests of Hypotheses • Two types of hypotheses: Null (H0)and alternative (H1)

36. Basic Ideas in Tests of Hypotheses • Set up H0 and H1. For a one-sided case, make sure these are set correctly. Usually these are done such that type 1 error becomes “costly” error. • Choose appropriate test statistic. This is usually based on the UMV estimator of the parameter under study. • Set up the decision rule if  = P(type 1 error) is specified. If not, report a p-value. • Choose a random sample and make the decision.

37. Setting up Hoand H1 • Suppose that the manufacturer of airbags for automobiles claims that the mean time to inflate airbag is no more than 0.1 second. • Suppose that the “costly error” is to conclude erroneously that the mean time is < 0.1. • How do we set up the hypotheses?

38. ILLUSTRATIVE EXAMPLE

40. Test on µ using normal • Sample size is large • Sample size is small, population is approximately normal with known .

41. DNR Region µ CP_1 CP_2

42. Computation of P(type 2 error)

43. Example (page 142) • µ = Mean propellant burning rate (in cm/s). • H0:µ = 50 vs H1:µ  50. • Two-sided hypotheses. • A sample of n=10 observations is used to test the hypotheses. • Suppose that we are given the decision rule. • Question 1: Compute P(type 1 error) • Question 2: Compute P(type 2 error when µ =52.

44. DECISION RULE

45. Calculation of P(type 1 error)

47. Example

48. Confidence Interval • Recall point estimate for the parameter under study. • For example, suppose that µ= mean tensile strength of a piece of wire. • If a random sample of size 36 yielded a mean of 242.4psi. • Can we attach any confidence to this value? • Answer: No! What do we do?

49. Confidence Interval (cont’d) • Given a parameter, say, , let denote its UMV estimator. • Given , 100(1-  )% CI for is constructed using the sampling (probability) distribution of as follows. • Find L and U such that P(L < < U) = 1- . • Note that L and U are functions of .