AP Statistics Curriculum 2007 Limits Poisson2Bin

From Socr

(Difference between revisions)
Jump to: navigation, search
m
Line 2: Line 2:
=== 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 Distirbution are contained here.]]
-
<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
+
-
===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 an appropriate when:
-
Models & strategies for solving the problem, data understanding & inference.  
+
: <math>np < 10</math>
-
 
+
: <math>n \geq 20</math> and <math>p \leq 0.05</math>.  
-
* 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 <math> n </math> is large and <math> p </math> is small with <math> np < 7 </math>This is true because
-
 
+
<math> \lim_{n \rightarrow \infty}
-
* TBD
+
{n \choose x} p^x(1-p)^{n-x}=\frac{\lambda^x e^{-\lambda}}{x!} </math>, where
-
   
+
<math> \lambda=np </math>.  Here is an example.  Suppose <math> 2\% </math> of a certain population have Type AB blood.  Suppose 60 people from this population are randomly selected.  The number of people <math> X </math> among the 60 that have Type AB blood follows the Binomial distribution with <math> n=60, p=0.02 </math>.  The figure below represents the distribution of <math> X </math>.  This figure also shows <math> P(X=0) </math>.
-
===Hands-on activities===
+
<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure13.jpg|600px]]</center>
-
Step-by-step practice problems.  
+
-
* TBD
+
* '''Note''': This distribution can be approximated well with Poisson with <math> \lambda=np=60(0.02)=1.2 </math>.  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 <math> X </math> which is about the same for both distributions).
 +
<center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure14.jpg|600px]]</center>
<hr>
<hr>
===References===
===References===
-
* TBD
 
<hr>
<hr>

Revision as of 01:33, 3 February 2008

Contents

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

Poisson as Approximation to Binomial Distribution

The complete details of the Poisson distribution as a limiting case of the Binomial Distirbution are contained here.

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 < 7. 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).

References




Translate this page:

(default)

Deutsch

Español

Français

Italiano

Português

日本語

България

الامارات العربية المتحدة

Suomi

इस भाषा में

Norge

한국어

中文

繁体中文

Русский

Nederlands

Ελληνικά

Hrvatska

Česká republika

Danmark

Polska

România

Sverige

Personal tools