Discrete Probability Distributions

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EMIS 7370 STAT 5340
Department of Engineering Management, Information
and Systems
Probability and Statistics for Scientists and
Engineers
Discrete Probability Distributions Hypergeometric
Poisson Distributions
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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Hypergeometric Distribution
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Hypergeometric Distribution

• Conditions
• Population size N
• K members are successes
• N - K members are failures
• Sample size n
• Obtained without replacement
• the number of successes in n trials

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Probability Mass Function
n if n ? k
x 0, 1, ...,
k if n gt k
where
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Mean Standard Deviation

• Mean or Expected Value of
• Standard Deviation of

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Hypergeometric Distribution
Example Five individuals from an animal
population thought to be near extinction in a
certain region have been caught, tagged and
released to mix into the population. After they
have had an opportunity to mix in, a random
sample of 10 of these animals is selected. Let
the number of tagged animals in the second
sample. If there are actually 25 animals of this
type in the region, Find a) b)
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Hypergeometric Distribution
Example Solution The parameter values are n
10 K 5 (5 tagged animals in the population) N
25
X 0, 1, 2, 3, 4, 5
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Hypergeometric Distribution
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Hypergeometric Distribution

• Similar to the Binomial Dist in Excel
• Click the Insert button on the menu bar (at the
  top of the Excel page)
• Go to the function option
• Choose Statistical from the Function Category
  window (a list of all available statistical
  functions will appear in the Function Name
  window)
• Choose the HYPGEOMDIST function
• Type in parameters
• Sample_s gt x
• Number_sample gt n
• Population_s gt k
• Number_pop gt N

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Poisson Distribution
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Properties
1. The number of outcomes occurring in one time
interval or specified region is independent of
the number that occurs in any other disjoint
time interval or region of space. In this way we
say that the Poisson process has no memory. 2.
The probability that a single outcome will occur
during a very short time interval or in a small
region is proportional to the length of the time
interval or the size of the region and does not
depend on the number of outcomes occurring
outside this time interval or region.
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Properties
3. The probability that more than one outcome
will occur in such a short time interval or fall
in such a small region is negligible. Remark
The Poisson distribution is used to describe a
number of processes, including the distribution
of telephone calls going through a switchboard
system, the demand (needs) of patients for
service at a health institution, the arrivals of
trucks and cars at a tollbooth, the number of
accidents at an intersection, etc.
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Poisson Distribution
Definition - If is the number of outcomes
occurring during a Poisson experiment, then
has a Poisson distribution with probability mass
function
where ? ?t and ? is the average number of
outcomes per unit time, t is the time interval
and e 2.71828...
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Poisson Distribution

• Mean or Expected Value of
• Variance and Standard Deviation of

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Poisson Distribution
Example When a company tests new tires by
driving them over difficult terrain, they find
that flat tires externally caused occur on the
average of once every 2000 miles. What is the
probability that in a given 500 mile test no more
than one flat will occur?
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Poisson Distribution
Example Solution Here the variable t is
distance, and the random variable of interest
is number of flats in 500 miles Since E(X)
is proportional to the time interval involved in
the definition of X, and since the average is
given as one flat is 2000 miles, we have
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Poisson Distribution
Example Solution The values assigned to ? and
?t depend on the unit of distance adopted. If we
take one mile as the unit, then t 500, ?
0.0005, and ? ?t 1/4. If we take 1000 miles
as the unit, then t 1/2, ? 1/2, and again ?t
1/4, and so on. The important thing is that ?t
1/4, no matter what unit is chosen.
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Poisson Distribution
Example Solution
Is this acceptable?