AP Statistics Curriculum 2007 Limits Norm2Poisson

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=== Normal Approximation to Poisson Distribution===
=== Normal Approximation to Poisson Distribution===
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Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
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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. 
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<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
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===Approach===
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For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=λ, \sigma^2=λ</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 [[continuity correction]] is performed, i.e., P(''X''&nbsp;≤&nbsp;''x''), where (lower-case) ''x'' is a non-negative integer, is replaced by P(''X''&nbsp;≤&nbsp;''x''&nbsp;+&nbsp;0.5).
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Models & strategies for solving the problem, data understanding & inference.  
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::  <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>
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* TBD
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===Model Validation===
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Checking/affirming underlying assumptions.  
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* TBD
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===Computational Resources: Internet-based SOCR Tools===
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* TBD
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===Examples===
===Examples===
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Computer simulations and real observed data.  
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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 62We can compute this as follows:
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<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.
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* TBD
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<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center>
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===Hands-on activities===
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Step-by-step practice problems.  
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* '''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 <math> P(54 \le X \le 62)=0.2718 </math>, which is close to the exact that was found earlier.  The figure below shows this probability.
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<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center>
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===References===
===References===
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* TBD
 
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Revision as of 01:51, 3 February 2008

Contents

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

Normal Approximation to Poisson Distribution

The Poisson(λ) distribution can be approximated with Normal when λ is large.

For sufficiently large values of λ, (say λ>1,000), the Normal(Failed to parse (lexing error): \mu=λ, \sigma^2=λ 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 continuity correction is performed, i.e., 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)\,

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 λ = 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




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