AP Statistics Curriculum 2007 Limits Norm2Poisson
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===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: | 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 [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability. | + | <math> P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. </math> The figure below from the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability. |
<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center> | <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center> | ||
Revision as of 17:00, 28 June 2010
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,σ^{2} = 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 x_{o} is a non-negative integer, and ), 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 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 the SOCR Poisson Distribution 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 using hte SOCR Normal Distribution Applet.
Problems
- SOCR Home page: http://www.socr.ucla.edu
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