# AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 04:55, 26 October 2009 (view source)IvoDinov (Talk | contribs) (ad)← Older edit Revision as of 16:19, 19 March 2010 (view source)IvoDinov (Talk | contribs) m (→Examples)Newer edit → Line 10: Line 10: ===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 $\lambda=50$.  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: 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. + $P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617.$  The figure below from [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability.
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]
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## 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 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 $N(\mu=50, \sigma=\sqrt{50}=7.071)$, 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.