# 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>. |

## Revision as of 18:20, 1 March 2008

## 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 SOCR 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 \( P(54 \le X \le 62)=0.2718 \), which is close to the exact that was found earlier. The figure below shows this probability.

### References

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

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