MATH408: Probability & StatisticsSummer 1999

WEEK 4

Dr. Srinivas R. ChakravarthyProfessor of Mathematics and Statistics

Kettering University(GMI Engineering & Management Institute)

Flint, MI 48504-4898Phone: 810.762.7906

Email: schakrav@kettering.eduHomepage: www.kettering.edu/~schakrav

Probability PlotExample 3.12

PROBABILITY MASS FUNCTION

Mean and variance of a discrete RV

Example 3.16

Verify that = 0.4 and = 0.6

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

BINOMIAL (cont’d)

Examples

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

Examples

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.

Examples

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.