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MATH408: Probability & Statistics Summer 1999 WEEK 4

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

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MATH408: Probability & Statistics Summer 1999 WEEK 4

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  1. MATH408: Probability & StatisticsSummer 1999WEEK 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 PlotExample 3.12

  3. PROBABILITY MASS FUNCTION

  4. Mean and variance of a discrete RV

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

  6. BINOMIAL RANDOM VARIABLE p defect Good 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

  7. BINOMIAL (cont’d)

  8. Examples

  9. 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.

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

  11. Graph of Poisson PMF

  12. Examples

  13. EXPONENTIAL DISTRIBUTION

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

  15. Examples

  16. 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.

  17. 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.

  18. Examples

  19. 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.

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