Lecture 13: Tue, Oct 22

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Lecture 13 Tue, Oct 22

• Announcements
• Exams, HW 4 back today
• HW 5 due Noon, this FRIDAY,
• in box in front of my office (JMHH 414-1).
• Today The Poisson Distribution
• Poisson experiment
• Probabilities, mean and variance
• Poisson approximation to the Binomial

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Discrete Distributions

• Today is our final discrete distribution
• General discrete (tables).
• The Binomial Distribution.
• The Poisson Distribution.

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Poisson Random Variables

• A Poisson random variable counts the number of
  events over time or space.
• Examples
• Number of phone calls received in a day.
• Number of defects on a sheet of metal.
• Number of taxis crossing a street corner in one
  minute.

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Requirements of a Poisson Experiment

• Independence. The number of occurrences in any
  interval is independent of the number of
  occurrences in any other interval.
• Homogeneity. The probability of an occurrence is
  the same for all intervals of the same size. The
  occurrence probability is proportional to the
  size of the interval.
• Rarity. The probability of two or more
  occurrences in an interval approaches zero as the
  interval becomes smaller.

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Poisson Events over Time
0hr
2hr
30min
1hr
Mean number of occurrences per hour
Mean over 30 minutes
Mean over 2 hours
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Poisson Events in a Spatial Region
Mean number of occurrences in 1 square foot
mu16
Mean for 4 square ft 4 Mean for 16 square ft 16
mu4
mu1
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Poisson Random Variable
A Poisson random variable counts the number of
occurrences in a Poisson experiment.
Event occurrence
0hr
2hr
1hr
X3 occurrences
X5 occurrences
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The Probability Distribution

• If X is a Poisson random variable with an
  occurrence rate , then its probability
  distribution function is
• where e 2.718.

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Example

• Accidents occur on a busy stretch of freeway at a
  rate of 1.5 per day. Find the probability that
  exactly one accident occurs in a day. The
  probability that 2 or fewer occur? 3 or more?

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Example where Intervals Dont Match

• Refer back to the last example. Find the
  probability of exactly 5 crashes over a 3-day
  period.

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Cumulative Probabilities

• Recall the definition of the cumulative
  probability distribution

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Using the Poisson Tables
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Poisson Tables
For the first example
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Poisson Probabilities in JMP-IN

• Assume XPoisson(mu3)
• Create 3 columns x, P(Xx), P(Xltx)
• Fill the x column from 0 to 15.
• Right click on column 2 header Formula/
  Probability/ Poisson Probability, and Poisson
  Probability(3,x)
• Right click on column 3 header Formula/
  Probability/ Poisson Distribution, and Poisson
  Distribution(3,x)

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Poisson Distribution,
Cumulative Probability Distribution
Probability Distribution
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Graphs of Some Poisson Distributions
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Quick Quiz

• Describe the shape of the distributions.
• What happens to the mean, median and mode as you
  increase
• What happens to the variance as you increase

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Mean and Variance

• The mean and variance of a Poisson random
  variable are the same!
• So the standard deviation is given by

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Poisson Approximation to the Binomial

• Binomial experiments
• Fixed number of trials n
• Constant success probability p
• Poisson experiments
• Infinite number of trials
• Constant success rate
• For n large and p small, Poisson provides an
  excellent approximation to the binomial if we set
  as the Poisson rate.

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Example of the Poisson/Binomial Approximation

• Consider two Binomial RVs.
• X Binomial (n25, p.04)
• X Binomial (n500, p.002)
• So both have mean 1.
• Which one will be better approximated by the
  Poisson?

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Example

• During the summer months (June to August,
  inclusive), an average of 5 marriages per month
  take place in a small city. Assuming that these
  marriages occur randomly and independently of
  each other, find the probability of the
  following
• Fewer than 4 marriages will occur in June.
• At least 14 but not more than 18 marriages will
  occur during the 3 months of summer.
• Exactly 10 marriages will occur during July and
  August.

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Example

• A snow-removal company bills its customers on a
  per-snowfall basis, rather than at a flat monthly
  rage. Based on the fee it charges per snowfall,
  the company will just break even in a month that
  has exactly six snowfalls. Suppose that the
  average number of snowfalls per month (during the
  winter) is eight.
• a) What is the probability that the company will
  just break even in a given winter month?
• b) What is the probability that the company will
  make a profit in a given winter month?

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Example

• Students arrive at the library to use a
  computer with a financial database. The arrival
  rate is 6 per hour. Assume the arrivals occur
  independently, and the arrival rate is constant.
• a) What is the probability that exactly two
  students will arrive in the next hour.
• b) What is the probability of 3 or fewer arrivals
  in a half hour period?
• c) What is the probability that the next person
  arrives within the next 10 minutes?

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Example

• Buses arrive at a bus stop according to a Poisson
  process at the rate of three buses per hour. The
  schedule is maintained every day at all hours.
• a) Find the chance that more than three buses
  arrive between 12 noon and 1pm.
• b) Find the chance that exactly three buses
  arrive between 12 noon and 1 pm, but no buses
  arrive between 1pm and 3pm.
• c) Find the chance that the seventh bus to arrive
  after 12 noon, arrives more than a full two hours
  after the sixth bus.

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Independent Poisson RVs

• Let X and Y be independent Poisson random
  variables with means
  respectively. Then XY has a Poisson
  distribution with mean

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Example

• There are two entrances to a parking garage.
  Cars come according to a Poisson distribution to
  the first entrance with a mean of 3 per minute.
  Cars come according to a Poisson distribution to
  the second entrance with a mean of 2 per minute.
  The number of arrivals to one entrance is
  independent of arrivals to the other entrance.
• a) What is the probability of at least two
  arrivals in a minute to the first entrance?
• b) What is the probability that the total number
  of arrivals (to both entrances) will be two or
  fewer in a one minute period?
• c) What are the mean and SD for the total number
  of arrivals in a one minute period?

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Example

• A machine produces defective items at a rate of 1
  per 50. If 100 items are randomly selected from
  the production line, what is the probability that
  more than 1 of them will be defective?