AP Statistics Curriculum 2007 Limits Norm2Poisson
From Socr
Line 2: | Line 2: | ||
=== Normal Approximation to Poisson Distribution=== | === Normal Approximation to Poisson Distribution=== | ||
- | + | 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=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'' ≤ ''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> | |
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
===Examples=== | ===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 <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 62. We can compute this as follows: | |
- | + | <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. | |
- | + | <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center> | |
- | + | ||
- | == | + | |
- | + | ||
- | * | + | * '''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. |
+ | <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center> | ||
<hr> | <hr> | ||
===References=== | ===References=== | ||
- | |||
<hr> | <hr> |
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).
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: 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 , together with the continuity correction for better approximation we obtain , 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
Translate this page: