# Difference between revisions of "AP Statistics Curriculum 2007 Limits Norm2Poisson"

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=== Normal Approximation to Poisson Distribution=== | === Normal Approximation to Poisson Distribution=== | ||

− | The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) | + | The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) Distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when <math> \lambda </math> is large. |

− | For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] | + | For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] Distribution is an excellent approximation to the [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(λ)]] Distribution. If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. |

If <math>x_o</math> is a non-negative integer, <math>X\sim Poisson(\lambda)</math> and <math>U\sim Normal(\mu=\lambda, \sigma^2=\lambda</math>), then <math>P_X(X<x_o) = P_U(U<x_o+0.5)</math>. | If <math>x_o</math> is a non-negative integer, <math>X\sim Poisson(\lambda)</math> and <math>U\sim Normal(\mu=\lambda, \sigma^2=\lambda</math>), then <math>P_X(X<x_o) = P_U(U<x_o+0.5)</math>. | ||

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===Examples=== | ===Examples=== | ||

Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with <math> \lambda=50 </math>. Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows: | Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with <math> \lambda=50 </math>. Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows: | ||

− | <math> P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. </math> The figure below from SOCR shows this probability. | + | <math> P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. </math> The figure below from the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability. |

<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center> | <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center> | ||

− | * '''Note''': We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain | + | * '''Note''': We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain $P(53.5<X<62.5| N(μ=50,σ=7.071))= 0.271759 $, which is close to the exact that was found earlier. The figure below shows this probability using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Applet]. |

<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center> | <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center> | ||

<hr> | <hr> | ||

− | === | + | |

+ | ===Applications: Positron Emission Tomography=== | ||

+ | The [http://en.wikipedia.org/wiki/Positron_emission_tomography physics of positron emission tomography (PET)] provides evidence that the Poisson distribution model may be used to study the process of radioactive decay using positron emission. As tracer isotopes decay, they give off positively charged electrons which collide with negatively charged electrons the result of which (by the law of preservation of energy) is one or a pair of photons emitted at the [http://en.wikipedia.org/wiki/Electron-positron_annihilation annihilation point] in space and detected by photo-multiplying tubes surrounding the imaged object (e.g., a human body part like the brain). The (large) number of arrivals at each detector is a Poisson process, which can be approximated by Normal distribution, as described above. This figure shows the schematics of the PET imaging technique. | ||

+ | <center>[[Image: SOCR_Activities_ExploreDistributions_Figure12.png|600px]]</center> | ||

+ | |||

+ | === See also=== | ||

+ | [[AP_Statistics_Curriculum_2007_Distrib_Poisson| Poisson Distribution Section of the EBook]]. | ||

+ | |||

+ | ===[[EBook_Problems_Limits_Norm2Poisson|Problems]]=== | ||

<hr> | <hr> | ||

* SOCR Home page: http://www.socr.ucla.edu | * SOCR Home page: http://www.socr.ucla.edu | ||

− | {{translate|pageName=http://wiki. | + | "{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Limits_Norm2Poisson}} |

## Latest revision as of 14:01, 3 March 2020

## Contents

## General Advance-Placement (AP) Statistics Curriculum - Normal Approximation to Poisson Distribution

### Normal Approximation to Poisson Distribution

The Poisson(\( \lambda \)) Distribution can be approximated with Normal when \( \lambda \) is large.

For sufficiently large values of λ, (say λ>1,000), the Normal(\(\mu=\lambda, \sigma^2=\lambda\)) Distribution is an excellent approximation to the Poisson(λ) Distribution. If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate continuity correction is performed.

If \(x_o\) is a non-negative integer, \(X\sim Poisson(\lambda)\) and \(U\sim Normal(\mu=\lambda, \sigma^2=\lambda\)), then \(P_X(X<x_o) = P_U(U<x_o+0.5)\).

### Examples

Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with \( \lambda=50 \). Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows\[ P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. \] The figure below from the SOCR Poisson Distribution shows this probability.

**Note**: We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using \( N(\mu=50, \sigma=\sqrt{50}=7.071) \), together with the continuity correction for better approximation we obtain $P(53.5<X<62.5| N(μ=50,σ=7.071))= 0.271759 $, which is close to the exact that was found earlier. The figure below shows this probability using the SOCR Normal Distribution Applet.

### Applications: Positron Emission Tomography

The physics of positron emission tomography (PET) provides evidence that the Poisson distribution model may be used to study the process of radioactive decay using positron emission. As tracer isotopes decay, they give off positively charged electrons which collide with negatively charged electrons the result of which (by the law of preservation of energy) is one or a pair of photons emitted at the annihilation point in space and detected by photo-multiplying tubes surrounding the imaged object (e.g., a human body part like the brain). The (large) number of arrivals at each detector is a Poisson process, which can be approximated by Normal distribution, as described above. This figure shows the schematics of the PET imaging technique.

### See also

Poisson Distribution Section of the EBook.

### Problems

- SOCR Home page: http://www.socr.ucla.edu

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