MATH408: Probability

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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_at_kettering.edu Homepag
e www.kettering.edu/schakrav
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Probability PlotExample 3.12
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PROBABILITY MASS FUNCTION
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Mean and variance of a discrete RV
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Example 3.16
Verify that ? 0.4 and ? 0.6
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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

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BINOMIAL (contd)
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Examples
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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.

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POISSON (contd)
If X is Poisson with parameter ?, then ? ? and
?2 ?
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Graph of Poisson PMF
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Examples
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EXPONENTIAL DISTRIBUTION
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MEMORYLESS PROPERTY

• P(X gt xy / X gt x) P( X gt y)
• ? X is exponentially distributed

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Examples
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Normal approximation to binomial(with correction
factor)

• Let X follow binomial with parameters n and p.
• P(X x) P( x-0.5 lt X lt x 0.5) and so we
  approximate this with a normal r.v with mean np
  and variance n p (1-p).
• GRT np gt 5 and n (1-p) gt 5.

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Normal approximation to Poisson (with correction
factor)

• Let X follow Poisson with parameter ?.
• P(X x) P( x-0.5 lt X lt x 0.5) and so we
  approximate this with a normal r.v with mean ?
  and variance ?.
• GRT ? gt 5.

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Examples
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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.