AP Statistics Curriculum 2007 Limits Poisson2Bin

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 Revision as of 18:54, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Current revision as of 17:00, 28 June 2010 (view source)Jenny (Talk | contribs) (5 intermediate revisions not shown) Line 1: Line 1: ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Poisson as Approximation to Binomial Distribution== ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Poisson as Approximation to Binomial Distribution== - === POisson as Approximation to Binomial Distribution=== + === Poisson as Approximation to Binomial 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_as_a_limiting_case_of_Binomial_distribution | complete details of the Poisson Distribution as a limiting case of the Binomial Distribution are contained here.]] -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ - ===Approach=== + * Note that the conditions of [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] approximation to [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial]] are complementary to the [[AP_Statistics_Curriculum_2007_Limits_Norm2Bin | conditions for Normal Approximation of Binomial Distribution]]. Poisson Approximation to Binomial is appropriate when: - Models & strategies for solving the problem, data understanding & inference. + : $np < 10$ - + : $n \geq 20 and [itex]p \leq 0.05$. - * TBD + - + - ===Model Validation=== + - Checking/affirming underlying assumptions. + - + - * TBD + - + - ===Computational Resources: Internet-based SOCR Tools=== + - * TBD + ===Examples=== ===Examples=== - Computer simulations and real observed data. + The [[About_pages_for_SOCR_Distributions | Binomial distribution]] can be approximated well by Poisson when $n$ is large and $p$ is small with $np < 10$, as stated above. This is true because + $\lim_{n \rightarrow \infty} + {n \choose x} p^x(1-p)^{n-x}=\frac{\lambda^x e^{-\lambda}}{x!}$, where + $\lambda=np$.  Here is an example.  Suppose $2\%$ of a certain population have Type AB blood.  Suppose 60 people from this population are randomly selected.  The number of people $X$ among the 60 that have Type AB blood follows the Binomial distribution with $n=60, p=0.02$.  The figure below represents the distribution of $X$.  This figure also shows $P(X=0)$. +
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure13.jpg|600px]]
- * TBD + * '''Note''': This distribution can be approximated well with Poisson with $\lambda=np=60(0.02)=1.2$.  The figure below is approximately the same as the figure above (the width of the bars is not important here.  The height of each bar represents the probability for each value of $X$ which is about the same for both distributions). - +
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure14.jpg|600px]]
- ===Hands-on activities=== + - Step-by-step practice problems. + - * TBD +
-
+ ===[[EBook_Problems_Limits_Poisson2Bin|Problems]]=== - ===References=== + - * TBD +

General Advance-Placement (AP) Statistics Curriculum - Poisson as Approximation to Binomial Distribution

Poisson as Approximation to Binomial Distribution

np < 10
$n \geq 20$ and $p \leq 0.05$.

Examples

The Binomial distribution can be approximated well by Poisson when n is large and p is small with np < 10, as stated above. This is true because $\lim_{n \rightarrow \infty} {n \choose x} p^x(1-p)^{n-x}=\frac{\lambda^x e^{-\lambda}}{x!}$, where λ = np. Here is an example. Suppose $2\%$ of a certain population have Type AB blood. Suppose 60 people from this population are randomly selected. The number of people X among the 60 that have Type AB blood follows the Binomial distribution with n = 60,p = 0.02. The figure below represents the distribution of X. This figure also shows P(X = 0).

• Note: This distribution can be approximated well with Poisson with λ = np = 60(0.02) = 1.2. The figure below is approximately the same as the figure above (the width of the bars is not important here. The height of each bar represents the probability for each value of X which is about the same for both distributions).