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 21:04, 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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