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What type of events does the Poisson distribution typically describe?
What type of events does the Poisson distribution typically describe?
In the context of the Poisson distribution, what does the random variable x represent?
In the context of the Poisson distribution, what does the random variable x represent?
Which of the following is a requirement for the Poisson distribution?
Which of the following is a requirement for the Poisson distribution?
What kind of intervals can a Poisson probability distribution be applied to?
What kind of intervals can a Poisson probability distribution be applied to?
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What is the correct formula component symbolized by 'e' in the Poisson distribution?
What is the correct formula component symbolized by 'e' in the Poisson distribution?
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Which of the following scenarios is likely to follow a Poisson distribution?
Which of the following scenarios is likely to follow a Poisson distribution?
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Who is credited with the development of the Poisson distribution?
Who is credited with the development of the Poisson distribution?
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Which of these examples would not typically be analyzed using a Poisson distribution?
Which of these examples would not typically be analyzed using a Poisson distribution?
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What is the means per day for motor vehicle deaths in Dutchess County, New York?
What is the means per day for motor vehicle deaths in Dutchess County, New York?
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Using a Poisson distribution, what would be the calculated probability of observing exactly 50 decayed radioactive atoms given a mean of 62.2274?
Using a Poisson distribution, what would be the calculated probability of observing exactly 50 decayed radioactive atoms given a mean of 62.2274?
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What is the probability of observing fewer than 3 truck arrivals in a two-hour period, given an average rate of 3 arrivals per hour?
What is the probability of observing fewer than 3 truck arrivals in a two-hour period, given an average rate of 3 arrivals per hour?
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If the mean arrival rate at a bus stop is 4.5 per quarter of an hour, what is the probability of exactly 0 arrivals?
If the mean arrival rate at a bus stop is 4.5 per quarter of an hour, what is the probability of exactly 0 arrivals?
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What is the mean number of occurrences in three time periods if the average is 2 per period in a Poisson distribution?
What is the mean number of occurrences in three time periods if the average is 2 per period in a Poisson distribution?
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How can a student compute the cumulative probability of having no more than 2 events occur in a given period for a Poisson distribution with a mean of 4.5?
How can a student compute the cumulative probability of having no more than 2 events occur in a given period for a Poisson distribution with a mean of 4.5?
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What is the result of calculating P(2) if the mean is 4.5 using the relevant Poisson function?
What is the result of calculating P(2) if the mean is 4.5 using the relevant Poisson function?
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What is the calculated probability of having more than 2 motor vehicle deaths on a typical day in Dutchess County?
What is the calculated probability of having more than 2 motor vehicle deaths on a typical day in Dutchess County?
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What parameter affects the Poisson distribution?
What parameter affects the Poisson distribution?
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In a binomial distribution, what are the possible values of the random variable x?
In a binomial distribution, what are the possible values of the random variable x?
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When can the Poisson distribution be used to approximate a binomial distribution?
When can the Poisson distribution be used to approximate a binomial distribution?
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What is the required condition for using the Poisson distribution as an approximation to the binomial distribution?
What is the required condition for using the Poisson distribution as an approximation to the binomial distribution?
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In the Illinois Pick 3 game example, what is the value of μ calculated as np?
In the Illinois Pick 3 game example, what is the value of μ calculated as np?
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What is the probability of winning exactly once in 365 days in the Illinois Pick 3 game using the Poisson distribution?
What is the probability of winning exactly once in 365 days in the Illinois Pick 3 game using the Poisson distribution?
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What is the probability of observing zero occurrences of a certain event as derived from the Poisson distribution in the provided example?
What is the probability of observing zero occurrences of a certain event as derived from the Poisson distribution in the provided example?
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What is the implication of the requirement np ≤ 10 in approximating a binomial distribution using the Poisson distribution?
What is the implication of the requirement np ≤ 10 in approximating a binomial distribution using the Poisson distribution?
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Which situation can be modeled by a Poisson distribution?
Which situation can be modeled by a Poisson distribution?
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Why is the number of duds found when testing four components not modeled by a Poisson distribution?
Why is the number of duds found when testing four components not modeled by a Poisson distribution?
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What is the probability of 4 arrivals in 30 minutes at Mercy Hospital, given an average rate of 6 per hour?
What is the probability of 4 arrivals in 30 minutes at Mercy Hospital, given an average rate of 6 per hour?
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In which scenario would the assumption of independent events likely fail?
In which scenario would the assumption of independent events likely fail?
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Why is the number of planes landing at O’Hare Airport between 8 and 9am not modeled by a Poisson distribution?
Why is the number of planes landing at O’Hare Airport between 8 and 9am not modeled by a Poisson distribution?
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How many radioactive atoms decayed in a day from a total of 1,000,000 after 365 days?
How many radioactive atoms decayed in a day from a total of 1,000,000 after 365 days?
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What characterizes events that can be modeled by a Poisson distribution?
What characterizes events that can be modeled by a Poisson distribution?
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In the context of a Poisson process, what does μ represent?
In the context of a Poisson process, what does μ represent?
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Study Notes
Poisson Probability Distributions
- Poisson distribution is a discrete probability distribution.
- It describes the behavior of rare events with small probabilities.
- Common occurrences are death of infants, misprints in a book, customer arrivals, or Geiger counter activations.
- The distribution was derived by Siméon Poisson in 1837.
- Initially used to describe the number of deaths by horse kicking in the Prussian Army.
- Other examples include patients arriving at an emergency room, radioactive decay, highway crashes, and internet users logging onto a website.
Learning Targets
- Poisson distribution is crucial for analyzing rare events.
Formula for Calculating Probability
- Formula for calculating probability: P(x)=e−μμxx!P(x) = \frac{e^{-\mu} \mu^x}{x!}P(x)=x!e−μμx
- Components:
- P(x): Probability of an event occurring x times in an interval.
- e: Approximately 2.71828.
- x: Number of occurrences of the event in the interval.
- μ: Mean, representing the average number of events in the given interval.
Requirements
- Random variable x is the number of occurrences of an event over some interval.
- Interval can be time, distance, area, volume, etc.
- Occurrences must be independent of each other.
- Occurrences must be uniformly distributed within the interval.
Parameters
- Mean (μ)
- Standard Deviation (σ = √μ)
Poisson or not?
- Determining if an event follows a Poisson distribution depends on whether the occurrences are independent and can be considered random within the time interval.
- Examples from the provided text.
Example: Mercy Hospital
- Patients arrive at the emergency room of Mercy Hospital at an average rate of 6 per hour on weekend evenings.
- Calculate the probability of 4 arrivals in 30 minutes (half an hour).
Example: Telecommunications
- Messages arrive at a switching center at an average rate of 1.2 per second.
- Calculate the probability of 5 messages arriving in a 2-second interval.
Example: Radioactive Atoms
-Radioactive atoms decay at a predictable rate.
- Calculate the daily average rate and the probability a specific number of radioactive atoms decay on a particular day.
Example: Trucks Arriving
- Trucks arrive at a receiving dock at an average rate of 3 per hour.
- Calculate the probability exactly 5 trucks will arrive in a two-hour period.
Example: Bus Arrivals
- Calculate the probability of fewer than 3 arrivals at a bus stop in a quarter of an hour.
Calculator Shortcut
- Using a TI-83+/84 calculator, use the poissonpdf and poissoncdf functions to obtain probabilities.
- poissonpdf (μ , x) = probability of exactly x successes
- poissoncdf (μ , x) = cumulative probability of 0 to x successes
Difference from a Binomial Distribution
- Binomial distribution depends on sample size (n) and probability (p).
- Poisson distribution only depends on the mean (μ).
- Binomial variables are limited to discrete values of 0, 1, ..., n
- Poisson variables have no upper limit (0, 1, 2, ...).
Condition of Poisson Distribution as an Approximation of Binomial Distribution
- A Binomial distribution can be approximated by a Poisson if the sample size (n) is large and the probability (p) of success is small.
Poisson Rule of Thumb
- The Poisson distribution can be used to approximate a binomial distribution, but requirements need to be met.
Examples
- Various examples concerning calculating probabilities using Poisson distribution in a variety of contexts.
Homework
- Exercises to practice using Binomial and Poisson distributions.
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Description
Test your understanding of Poisson distributions, a key concept in probability theory that describes the likelihood of rare events. This quiz covers the behavior, applications, and calculation formulas associated with Poisson distributions. Challenge yourself with various scenarios related to this statistical model.
