Transcript MATH408: PROBABILITY & STATISTICS

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]
Homepage: www.kettering.edu/~schakrav
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 (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.