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

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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. | 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>]] distribution is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [ | + | 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. Suppose P(''X''≤''x''), where (lower-case) ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5). |

:: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math> | :: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math> | ||

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

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

## Revision as of 22:04, 2 February 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. Suppose P(*X*≤*x*), where (lower-case) *x* is a non-negative integer, is replaced by P(*X* ≤ *x* + 0.5).

- \[F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,\]

If \(X\sim Poisson(\lambda)\) and \(U\sim Normal(<math>\mu=\lambda, \sigma^2=\lambda\)), 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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