Math408 probability statistics summer 1999 week 4
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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: [email protected]

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MATH408: Probability & Statistics Summer 1999 WEEK 4

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Math408 probability statistics summer 1999 week 4

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: [email protected]

Homepage: www.kettering.edu/~schakrav


Probability plot example 3 12

Probability PlotExample 3.12


Math408 probability statistics summer 1999 week 4

PROBABILITY MASS FUNCTION


Mean and variance of a discrete rv

Mean and variance of a discrete RV


Example 3 16

Example 3.16

Verify that  = 0.4 and  = 0.6


Math408 probability statistics summer 1999 week 4

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


Math408 probability statistics summer 1999 week 4

BINOMIAL (cont’d)


Examples

Examples


Poisson random variable

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.


Math408 probability statistics summer 1999 week 4

POISSON (cont’d)

If X is Poisson with parameter , then  =  and 2 = 


Math408 probability statistics summer 1999 week 4

Graph of Poisson PMF


Examples1

Examples


Math408 probability statistics summer 1999 week 4

EXPONENTIAL DISTRIBUTION


Memoryless property

MEMORYLESS PROPERTY

P(X > x+y / X > x) = P( X > y)

 X is exponentially distributed


Examples2

Examples


Normal approximation to binomial with correction factor

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.


Normal approximation to poisson with correction factor

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.


Examples3

Examples


Home work problems use minitab

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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