1. MATH408: Probability & Statistics Summer 1999 WEEK 4 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. Probability Plot Example 3.12

6. PROBABILITY MASS FUNCTION

7. Mean and variance of a discrete RV

8. Example 3.16 Verify that  = 0.4 and  = 0.6

9. BINOMIAL RANDOM VARIABLE defect Good p q n, items are sampled, is fixed P(defect) = p is the same for all independently and randomly chosen X = # of defects out of n sampled

10. BINOMIAL (cont’d)

12. Examples

13. POISSON RANDOM VARIABLE Named after Simeon D. Poisson (1781-1840) Originated as an approximation to binomial Used extensively in stochastic modeling Examples include: –Number of phone calls received, number of messages arriving at a sending node, number of radioactive disintegration, number of misprints found a printed page, number of defects found on sheet of processed metal, number of blood cells counts, etc.

14. POISSON (cont’d) If X is Poisson with parameter, then  = and  2 =

15. Graph of Poisson PMF

16. Examples

17. EXPONENTIAL DISTRIBUTION

19. MEMORYLESS PROPERTY P(X > x+y / X > x) = P( X > y)  X is exponentially distributed

20. Examples

21. Normal approximation to binomial (with correction factor) Let X follow binomial with parameters n and p. P(X = x) = P( x-0.5 < X < x + 0.5) and so we approximate this with a normal r.v with mean np and variance n p (1-p). GRT: np > 5 and n (1-p) > 5.

22. Normal approximation to Poisson (with correction factor) Let X follow Poisson with parameter. P(X = x) = P( x-0.5 < X < x + 0.5) and so we approximate this with a normal r.v with mean and variance. GRT: > 5.

23. Examples

24. HOME WORK PROBLEMS (use Minitab) Sections: 3.6 through 3.10 51, 54, 55, 58-60, 61-66, 70, 74-77, 79, 81, 83, 87-90, 93, 95, 100-105, 108 Group Assignment: (Due: 4/21/99) Hand in your solutions along with MINITAB output, to Problems 3.51 and 3.54.