AP Statistics Curriculum 2007 Limits Norm2Poisson

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(Normal Approximation to Poisson Distribution)
(Normal Approximation to Poisson Distribution)
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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.   
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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''&nbsp;≤&nbsp;''x''&nbsp;+&nbsp;0.5).
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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.  
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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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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>.
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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>.
===Examples===
===Examples===

Revision as of 01:57, 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(μ = λ,σ2 = λ) 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 xo is a non-negative integer, X\sim Poisson(\lambda) and U\sim Normal(\mu=\lambda, \sigma^2=\lambda), then PX(X < xo) = PU(U < xo + 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 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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