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MATH408: Probability & Statistics Summer 1999 WEEK 4

This document summarizes key concepts from Week 4 of the Probability and Statistics course, taught by Dr. Srinivas R. Chakravarthy at Kettering University. It covers probability mass functions, binomial and Poisson random variables, exponential distributions, and the normal approximation of binomial and Poisson distributions. The lecture notes include examples, definitions, properties, and homework problems that utilize Minitab for statistical analysis. Understanding these topics is essential for mastering discrete random variables and their applications.

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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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