MATH408: Probability & Statistics Summer 1999 WEEK 4 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI Phone: Homepage:

Probability Plot Example 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 ( ) 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 , 54, 55, 58-60, 61-66, 70, 74-77, 79, 81, 83, 87-90, 93, 95, , 108 Group Assignment: (Due: 4/21/99) Hand in your solutions along with MINITAB output, to Problems 3.51 and 3.54.