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=== Poisson Random Variables and Experiments===
 
=== Poisson Random Variables and Experiments===
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*Definition: '''Poisson distribution''' is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
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*Definition: '''Poisson Distribution''' is a discrete probability distribution that expresses the probability of the number of events occurring in a fixed period of time. These events occur with a known average rate and are independent of the time since the last event. The Poisson Distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
  
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*Mass function: For '''X~Poisson(λ)''', the Poisson mass function is given by <math>P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!</math>, where
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*Mass function: For '''X~Poisson(λ)''', the Poisson mass function is given by <math>P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!</math> where
 
** ''e'' is the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number] (''e'' = 2.71828...)
 
** ''e'' is the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number] (''e'' = 2.71828...)
 
** ''k'' is the number of occurrences of an event - the probability of which is given by the mass function
 
** ''k'' is the number of occurrences of an event - the probability of which is given by the mass function
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* Notes
 
* Notes
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** The Poisson distribution can be derived as a limiting case of the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | binomial distribution]].
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** The [[AP_Statistics_Curriculum_2007_Distrib_Poisson#Poisson_as_a_Limiting_Case_of_Binomial_Distribution|Poisson Distribution can be derived as a limiting case]] of the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial Distribution]].
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** The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms and [http://en.wikipedia.org/wiki/Positron_emission_tomography Positron Emission Tomography imaging].   
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** The Poisson Distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms and [http://en.wikipedia.org/wiki/Positron_emission_tomography Positron Emission Tomography Imaging].   
  
 
* Expectation: The [[AP_Statistics_Curriculum_2007_Distrib_MeanVar | expected value]] of a Poisson distributed random variable ''X'' is λ.
 
* Expectation: The [[AP_Statistics_Curriculum_2007_Distrib_MeanVar | expected value]] of a Poisson distributed random variable ''X'' is λ.
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*Example: See [[SOCR_EduMaterials_Activities_PoissonExperiment | this SOCR Poisson distribution activity]].
 
*Example: See [[SOCR_EduMaterials_Activities_PoissonExperiment | this SOCR Poisson distribution activity]].
  
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===Examples===
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===Example===
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The Poisson distribution arises in many different situations of discrete nature when the probability of the phenomenon happening is constant in time or space. Examples of events that may be modeled by Poisson distribution include:
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Suppose one expects to see 40 shooting stars per hour at night (by observing at a specific location and not using visual enhancers).  Assume that the process of observing arrivals (shooting stars) is <math>Poisson(\lambda=40)</math>. 
 +
 
 +
* Compute the probability that in a 1 hour timeframe, the number of observed shooting stars will be between 44 and 52! We can compute this as follows: Let <math>X\sim Poisson(\lambda)</math>.
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<math> P(44 \le X \le 52) = \sum_{x=44}^{52} \frac{40^x e^{-40}}{x!}=0.255719.</math>  The figure below shows this result using [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions]:
 +
<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
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 +
* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P\big (43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325) \big )=0.265947$, which is close to the exact probability ($0.255719$) that was found earlier.  The figure below shows this probability.
 +
<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>
 +
 
 +
* More about [[SOCR_EduMaterials_Activities_Explore_Distributions |approximation of Poisson by Normal distribution is available here]].
 +
 
 +
===Applications===
 +
The Poisson Distribution arises in many different discrete situations when the probability of the observed phenomenon is constant in time or space. Examples of events that may be modeled by Poisson Distribution include:
 
* The number of cars that pass through a certain point on a road (sufficiently distant from traffic lights) during a given period of time.
 
* The number of cars that pass through a certain point on a road (sufficiently distant from traffic lights) during a given period of time.
 
* The number of spelling mistakes one makes while typing a single page.
 
* The number of spelling mistakes one makes while typing a single page.
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* The number of particles that "scatter" off of a target in a nuclear or high energy physics experiment.
 
* The number of particles that "scatter" off of a target in a nuclear or high energy physics experiment.
  
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===Poisson as a limiting case of Binomial distribution===
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===Poisson as a Limiting Case of Binomial Distribution===
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In several of the above examples, the events being counted are actually the outcomes of discrete trials, and would more precisely be modeled using the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial distribution]].  However, the binomial distribution with parameters ''n'' (number of trials) and <math>p={λ\over n}</math> (probability of success). That is, Binomial approaches the Poisson distribution with expected value <math>\lambda=p\times n</math>, as ''n'' approaches infinity. It provides a means to approximate Binomial random variables (for large n) using the Poisson distribution. The details of this approximation are below:
+
In several of the above examples, the events being counted are actually the outcomes of discrete trials, and would more precisely be modeled using the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial Distribution]] with parameters ''n'' (number of trials) and <math>p={\lambda \over n}</math> (probability of success). That is, Binomial(n,p) approaches the Poisson(<math>\lambda</math>) distribution, where expected value <math>\lambda=p\times n</math>, as ''n'' approaches infinity. This provides the means to approximate Binomial random variables (for large n) using the Poisson distribution. The details of this approximation are below:
  
 
* <math>\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda}.</math>
 
* <math>\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda}.</math>
  
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Let ''p'' = λ/''n''.  Then we have
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Let <math>p = \frac{\lambda}{n}</math> and <math>X \sim Binomial(n, p)</math>.  Then we have
  
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:<math>\lim_{n\to\infty} \Pr(X=k)=\lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k}
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:<math>\lim_{n\to\infty} P(X=k)=\lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k}
 
=\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}</math>
 
=\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}</math>
  
 
:<math>=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)}\ \underbrace{\left({\lambda^k \over k!}\right)}\ \underbrace{\left(1-{\lambda \over n}\right)^n}\ \underbrace{\left(1-{\lambda \over n}\right)^{-k}}</math>
 
:<math>=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)}\ \underbrace{\left({\lambda^k \over k!}\right)}\ \underbrace{\left(1-{\lambda \over n}\right)^n}\ \underbrace{\left(1-{\lambda \over n}\right)^{-k}}</math>
  
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* As ''n'' approaches ∞, the first term approaches 1; the second remains constant since "''n''" does not appear in it at all; the third approaches ''e''<sup>-λ</sup>; and the fourth expression approaches 1.
+
* As <math>n \longrightarrow \infty</math> the first term approaches 1; the second remains constant since "''n''" does not appear in it at all; the third approaches ''e''<sup>-λ</sup>; and the fourth expression approaches 1.
  
 
* Consequently the limit is
 
* Consequently the limit is
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* More generally, whenever a sequence of binomial random variables with parameters ''n'' and ''p''<sub>''n''</sub> is such that
 
* More generally, whenever a sequence of binomial random variables with parameters ''n'' and ''p''<sub>''n''</sub> is such that
 
:<math>\lim_{n\rightarrow\infty} np_n = \lambda,</math>
 
:<math>\lim_{n\rightarrow\infty} np_n = \lambda,</math>
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the sequence convergence (in distribution) to a Poisson random variable with mean λ.
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the sequence converges  (in distribution) to a Poisson random variable with mean λ.
 +
 
 +
===See also===
 +
[[AP_Statistics_Curriculum_2007_Limits_Norm2Poisson| Normal Approximation to Poisson distribution and positron emission tomography (PET) applications]].
 +
 
 +
===[[EBook_Problems_Distrib_Poisson|Problems]]===
  
 
===References===
 
===References===
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* SOCR Home page: http://www.socr.ucla.edu
 
* SOCR Home page: http://www.socr.ucla.edu
  
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Poisson}}
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"{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Poisson}}

Latest revision as of 14:37, 3 March 2020

General Advance-Placement (AP) Statistics Curriculum - Poisson Random Variables and Experiments

Poisson Random Variables and Experiments

  • Definition: Poisson Distribution is a discrete probability distribution that expresses the probability of the number of events occurring in a fixed period of time. These events occur with a known average rate and are independent of the time since the last event. The Poisson Distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
  • Mass function: For X~Poisson(λ), the Poisson mass function is given by \(P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!\) where
    • e is the natural number (e = 2.71828...)
    • k is the number of occurrences of an event - the probability of which is given by the mass function
    • \(k! = 1\times 2\times 3\times \cdots \times k\)
    • λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ=10/4=2.5.
  • Expectation: The expected value of a Poisson distributed random variable X is λ.

Example

Suppose one expects to see 40 shooting stars per hour at night (by observing at a specific location and not using visual enhancers). Assume that the process of observing arrivals (shooting stars) is \(Poisson(\lambda=40)\).

  • Compute the probability that in a 1 hour timeframe, the number of observed shooting stars will be between 44 and 52! We can compute this as follows: Let \(X\sim Poisson(\lambda)\).

\( P(44 \le X \le 52) = \sum_{x=44}^{52} \frac{40^x e^{-40}}{x!}=0.255719.\) The figure below shows this result using SOCR distributions:

SOCR EBook Dinov RV Poisson 013008 Fig7.jpg
  • Note: We observe that this distribution is bell-shaped. We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate continuity correction is performed. Using \(Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)\), together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P\big (43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325) \big )=0.265947$, which is close to the exact probability ($0.255719$) that was found earlier. The figure below shows this probability.
SOCR EBook Dinov RV Poisson 013008 Fig8.jpg

Applications

The Poisson Distribution arises in many different discrete situations when the probability of the observed phenomenon is constant in time or space. Examples of events that may be modeled by Poisson Distribution include:

  • The number of cars that pass through a certain point on a road (sufficiently distant from traffic lights) during a given period of time.
  • The number of spelling mistakes one makes while typing a single page.
  • The number of phone calls at a call center per minute.
  • The number of times a web server is accessed per minute.
  • The number of road kill (animals killed) found per unit length of road.
  • The number of mutations in a given stretch of DNA after a certain amount of radiation exposure.
  • The number of unstable atomic nuclei that decayed within a given period of time in a piece of radioactive substance.
  • The number of pine trees per unit area of mixed forest.
  • The number of stars in a given volume of space.
  • The distribution of visual receptor cells in the retina of the human eye.
  • The number of light bulbs that burn out in a certain amount of time.
  • The number of viruses that can infect a cell in cell culture.
  • The number of hematopoietic stem cells in a sample of unfractionated bone marrow cells.
  • The number of inventions of an inventor over their career.
  • The number of particles that "scatter" off of a target in a nuclear or high energy physics experiment.

Poisson as a Limiting Case of Binomial Distribution

In several of the above examples, the events being counted are actually the outcomes of discrete trials, and would more precisely be modeled using the Binomial Distribution with parameters n (number of trials) and \(p={\lambda \over n}\) (probability of success). That is, Binomial(n,p) approaches the Poisson(\(\lambda\)) distribution, where expected value \(\lambda=p\times n\), as n approaches infinity. This provides the means to approximate Binomial random variables (for large n) using the Poisson distribution. The details of this approximation are below:

  • \(\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda}.\)

Let \(p = \frac{\lambda}{n}\) and \(X \sim Binomial(n, p)\). Then we have

\[\lim_{n\to\infty} P(X=k)=\lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k} =\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}\]

\[=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)}\ \underbrace{\left({\lambda^k \over k!}\right)}\ \underbrace{\left(1-{\lambda \over n}\right)^n}\ \underbrace{\left(1-{\lambda \over n}\right)^{-k}}\]

  • As \(n \longrightarrow \infty\) the first term approaches 1; the second remains constant since "n" does not appear in it at all; the third approaches e-λ; and the fourth expression approaches 1.
  • Consequently the limit is

\[{\lambda^k e^{-\lambda} \over k!}.\,\!\]

  • More generally, whenever a sequence of binomial random variables with parameters n and pn is such that

\[\lim_{n\rightarrow\infty} np_n = \lambda,\] the sequence converges (in distribution) to a Poisson random variable with mean λ.

See also

Normal Approximation to Poisson distribution and positron emission tomography (PET) applications.

Problems

References

"-----

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