# AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 18:56, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 01:53, 3 February 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 2: Line 2: === Normal Approximation to Poisson Distribution=== === Normal Approximation to Poisson Distribution=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson( \lambda [/itex]) distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when $\lambda is large. - [[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]] + - ===Approach=== + For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal([itex]\mu=λ, \sigma^2=λ$]] 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). - Models & strategies for solving the problem, data understanding & inference. + ::  $F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$ - + - * TBD + - + - ===Model Validation=== + - Checking/affirming underlying assumptions. + - + - * TBD + - + - ===Computational Resources: Internet-based SOCR Tools=== + - * TBD + ===Examples=== ===Examples=== - Computer simulations and real observed data. + 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. - * TBD +
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]
- + - ===Hands-on activities=== + - Step-by-step practice problems. + - * TBD + * '''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. +
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]

===References=== ===References=== - * TBD

## 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).

$F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,$

### 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.