Statistical Processes ENMA 420/520

1. Statistical ProcessesENMA 420/520 Michael F. Cochrane, Ph.D. Dept. of Engineering Management And Systems Engineering Old Dominion University

2. Class SixReadings & Problems • Reading assignment • M & S • Chapter 6 Sections 6.9 - 6.10 • Berk and Carey Chapter 6 • Recommended problems • M & S Chapter 6 • 100, 101, 102, 105, 106, 120, 121

3. Sampling DistributionsFoundation of Inferential Statistics • So far discussed • Issues concerning sampling • Random sampling • From a known population • From a population modeled as a rv • Concept of a sampling distribution • Probability distribution of a statistic • Standard error of a statistic • Why does it arise?

4. What is this? Sampling DistributionTopics • Distribution of a function of rv’s • Can I determine the distribution of g(y1, …, yn)? • Sampling from a rv • How do I randomly sample from a rv representing a population? • The Central Limit Theorem • Keystone to inferential statistics • What happens when I sum many iid rv’s? • Other sampling distributions • Other useful sampling distributions

5. PDF of a function of RVsA Messy Calculation • Trying to find pdf of w, where w = g(y1, …, yn) yi are all random variables • Recall previous demo of @Risk & discussion • If w = sum of ~N rv’s • Then w ~N • If w = y1 + y2, where yi~U, • Then w~Triangular • Can we in general find pdf of w? • Section 6.6 explains one method • Messy except for simple examples! Suppose w = y1 + y2 where, y1~N & y 2~ What is the pdf of w? What is E(w)? What is the VAR(w)?

6. PDFs of Functions of RVsWhat’s Next • Example 6.13 of text laborious • Work your way through math • Note analytical result • If w = y1 + y2, where yi~U, • Then w~Triangular • Next apply same concept to find random sample of a rv Same as result using @Risk

7. Generating Random SampleInverse Transformation Method Method generates random samples from rv y, given that CDF of y is known • Steps • Generate F(y), graphically or analytically • Generate n random samples of w, where w~U(0,1) • Solve yi = F-1(wi), for i=1, 2, …, n • Set of yi is set of random observations from y

8. Solve for yi: Example Problem Generating from Exponential RV Given y~Exp(beta=5), find 3 random observations from y Repeat process for w2 and w3

9. Generating n random numbers in range [0,1] wi yi Finding corresponding yi Using inverse function y=F-1(w) OR graphically What Are We Doing in This Method? F(y) 1

10. Sum of n iid random variables Sums of Random VariablesSomething Interesting Occurs • Previously discussed sum w = x + y • Where x & y are random variables • Convenient result when x & y are ~N • Result not intuitive when x & y are ~U • Recall w then ~Triang • Lets investigate what happens to pdf of w • Summation of n random variables yi • For varying n Use @Risk add-in to simulate results. Remember: Analytical solution very difficult - resort to simulation.

11. @Risk Demo

12. For yi iid random variables What Happens When You Sum RVs?You Don’t Get a Trailer Park! • Seems like w starts looking like a normal distribution as n increases • Formalize as the Central Limit Theorem Let y1, y2, …, yn be iid random variables, and w = lim n y1 + y2 + … + yn THEN w ~ N • In practice w~N for n >25 to 30

13. What does represent? For n sufficiently large, what is the distribution of ? Expanding On the CLTCan We Use Results for Sampling? • From CLT know that w~N • How can we use CLT for sampling? • Let’s construct sampling distribution of sample mean Why is it a rv?

14. Population y Pay close attention to these!! The Sampling Distribution of the MeanA Very Convenient Result! • CLT allows us to say ~N • What are are assumptions about distribution of y? • What is assumption about sample size? Many samples of n observations y y2

15. What is the mean? Variance? Be sure to understand the distinction between these & their relationship! What is the mean? Variance? ComparisonPDFs of y vs. y_bar What happens to the variance if you increase n?

16. For yi ~N random variables Then w~N VAR(w) = w2 = 12 + 22 + … + n2 + 2 122 + … + 2 1n2 + … + 2 (n-1)n2 What are the ij2 in the above? Extension to CLTSumming Normally Distributed RVs • Theorem 6.10 extends the CLT to a special case Note: assumption of iid random variables NOT THERE! E(w) = E(y1) + E(y2) + … + E(yn) What if all yi are independent?

17. Perform study and found y~Exp() y = time to perform corrective maintenance  = 1700 hours Sample 70 jobs, what is p(mean of sample >2500 hours)? If you find mean of sample > 2500 hrs what would you conclude? Population y~Exp() Example Problem (prob. 6.72, p. 245) Sample of 70 observations y=  =1700 y2= 2

18. Distribution of y What do the PDFs Look Like?Of y and y_bar

19. What do you suspect the probability is? Where do the terms here come from? Part aLet’s Do the Math P(z>3.94)  0 Part b What would you suspect about y if mean of sample 2500?

20. Group Exercise • Elevator can carry maximum of 10,000 lbs. • Load contains 45 boxes each of weigh y • y is a rv, with  = 200 lbs.,  = 55 lbs. • Find probability that all 45 boxes can be simultaneously carried on the elevator?

21. This is the exact probability can we use w to approximate this probability??? Approximating DistributionsBased on Sum of RVs • Recall binomial distribution w = y1 + y2 + … + yn w is rv - number of successes in n trials What happens to distribution of w as n   ?? Not practical for n   , can we use results just when n becomes large? What is the E(w)?? What is the VAR(w)??

22. What do these represent? Continuity Correction factor Approximating Binomial DistributionRemember Question on Quiz #1 Let y~Bin(n,p), then can approximate using a normal distribution What are the denominators?

23. Sample ProblemApproximating the Binomial Have shipment of 2800 watermelons, of which 25% estimated to be green • Choose sample of 50, find probability that between 12 to 14 (inclusive) are green • Set up problem • To solve exactly • To solve using normal approximation

24. Here’s The Exact Solution • Now solve the problem using the normal approximation method

25. Normal Approximation Problem

26. Binomial & Normal Look Similar One Discrete the Other Continuous

27. When Is Approximation Appropriate? • 0 <   2 < n • Why??? • No reason to use with Excel available! • Understanding concepts provides insight into statistics

28. A Couple of More Sampling Distributions • Given random sample of n observations from rv y, where y ~ N(, 2) • What is the distribution of ? • Theorem 6.11 says that distribution of s2 is not normally distributed Note this KEY assumption for purposes of distribution of s2 What is the random variable s2 ??? Note that it may not even be symmetric. What distribution is it?

29. Which of these terms are random variables? And, what do they represent? Distribution of s2A Scaled 2 Distribution What is the E(2 )? What would you expect E(s2) to be?

30. Expected value of s2

31. Population y y y Let’s Review What Is Happening Many samples of n observations s2 s^2 s^2 Note that in this case we are describing the pdf of s2 and NOT that of y_bar!

32. Take note! Example Time to solve a problem is ~N(0.8, 2.25) • Q: take a sample of 30 observations, find p(s2>3.30) y_bar~N(0.8,.2738) 5.9 2.9 0.4 0.6 0.7 0.9 1.0 1.2 0.0 Can you explain N(0.8, 0.2738)? Can you explain X2(29)?

33. Using Excel CHIDIST(42.533, 29) Working Out Problem • What is problem asking for: • p(s2>3.30) • Solve by setting this equal to • p(s2 > 3.30) = p(2 > 2 0) • Now need to find • 2 0 • p(2 > 2 0)

34. Last Comment About 2As  Increases (past 30) 2~N Hint as to why: recall, 2 = z12 + z22 + … + z2

35. Class SixReadings & Problems • Reading assignment • M & S • Chapter 6 Sections 6.9 - 6.10 • Berk and Carey Chapter 6 • Recommended problems • M & S Chapter 6 • 100, 101, 102, 105, 106, 120, 121