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# MATH408: Probability & Statistics Summer 1999 WEEK 4 PowerPoint PPT Presentation

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

MATH408: Probability & Statistics Summer 1999 WEEK 4

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### 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

### Probability PlotExample 3.12

PROBABILITY MASS FUNCTION

### Example 3.16

Verify that  = 0.4 and  = 0.6

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

BINOMIAL (cont’d)

### 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.

POISSON (cont’d)

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

Graph of Poisson PMF

### Examples

EXPONENTIAL DISTRIBUTION

### MEMORYLESS PROPERTY

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

 X is exponentially distributed

### 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)

• 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.

### 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.