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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: [email protected]: 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.