SOCR BivariateNormal JS Activity - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T16:41:01Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

IvoDinov: /* References */

2013-05-01T17:15:14Z

‎References

IvoDinov: typo

2012-08-22T02:40:46Z

typo

IvoDinov: /* Parameter Estimation */

2012-08-16T22:36:54Z

‎Parameter Estimation

IvoDinov: /* Background */

2012-08-16T18:49:25Z

‎Background

IvoDinov: /* Inflation vs. HPI */

2012-07-30T20:47:53Z

‎Inflation vs. HPI

IvoDinov: /* Learning Activity: Human Height and Weight */

2012-07-30T20:44:37Z

‎Learning Activity: Human Height and Weight

IvoDinov: /* Learning Activity: Human Height and Weight */

2012-07-30T20:31:31Z

‎Learning Activity: Human Height and Weight

IvoDinov at 20:24, 30 July 2012

2012-07-30T20:24:29Z

IvoDinov: /* Experiment 2: Inflation vs. HPI */

2012-07-25T01:03:22Z

‎Experiment 2: Inflation vs. HPI

← Older revision		Revision as of 16:41, 3 March 2020
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 ==References==
 * [[AP_Statistics_Curriculum_2007_MultivariateNormal|See the EBook Multivariate Normal Distribution Chapter]]
 * Dinov, ID, Christou, N and Sanchez, J. (2008) [http://www.amstat.org/publications/jse/v16n2/dinov.html Central Limit Theorem: New SOCR Applet and Demonstration Activity], [http://www.amstat.org/publications/jse/ Journal of Statistics Education], Volume 16, Number 2.
 {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_BivariateNormal_JS_Activity}}

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 * Provide a visualization tool for better understanding of the bivariate normal distribution.
 * Clarify the definitions and interplay between marginal, conditional and joint probability distributions (in the bivariate Normal case).
-* To learn how to calculate Normal marginal, conditional and joint probabilities.
+* Learn how to calculate Normal marginal, conditional and joint probabilities.
 * Show how correlation influences the distribution of two normally-distributed variables. Demonstrate that when X and Y have joint bivariate normal distribution with zero correlation, then X and Y must be independent.
 * Build a framework for generalizing the univariate normal distribution to higher dimensions.

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 <center>[[Image:SOCR_BivariateNormal_JS_Activity_Fig4.png|300px]]</center>
-* To estimate the '''correlation''' between height and weight we use the [http://www.socr.ucla.edu/htmls/ana/SimpleRegression_Analysis.html SOCR SLR applet], see the corresponding [[SOCR_EduMaterials_AnalysisActivities_SLR|activity]]. Map height as the dependent variable and weight as the independent variable. Click “Calculate” to attain the summary statistics. The correlation between Height and Weight is 0.557.
+* To estimate the '''correlation''' between height and weight we use the [http://www.socr.ucla.edu/htmls/ana/SimpleRegression_Analysis.html SOCR SLR applet], see the corresponding [[SOCR_EduMaterials_AnalysisActivities_SLR|activity]]. Map height as the dependent variable and weight as the independent variable. Click “Calculate” to obtain the summary statistics. The correlation between Height and Weight is 0.557.
 <center>[[Image:SOCR_BivariateNormal_JS_Activity_Fig5.png|300px]]</center>

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 : 3. The '''joint probability density function of \(X\) and \(Y\)''' is:
 :: \( f_{X,Y}(x,y) = g_{X|Y}(x|y)f_Y(y) = \frac{1}{\sigma_X\sigma_Y 2\pi\sqrt{1-\rho^2}} e^{-q(x,y)} \), where
-::: \( q(x,y) =  \frac{1}{2} \frac{1}{1-\rho^2} \left ( \left (  \frac{X-\mu_X}{\sigma_X}  \right )^2 -2\rho\frac{X-\mu_X}{\sigma_X}\frac{Y-\mu_Y}{\sigma_Y} +\left (  \frac{Y-\mu_Y}{\sigma_Y}  \right )^2 \right ) \).
+::: \( q(x,y) =  \frac{1}{2} \frac{1}{1-\rho^2} \left ( \left (  \frac{x-\mu_X}{\sigma_X}  \right )^2 -2\rho\frac{x-\mu_X}{\sigma_X}\frac{y-\mu_Y}{\sigma_Y} +\left (  \frac{y-\mu_Y}{\sigma_Y}  \right )^2 \right ) \).
 ==Requirements & usability==

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 Use the [[SOCR_Data_MonetaryBaseStocksInterest1959_2009|SOCR Inflation vs. Housing Price Index (HPI) dataset]].
 * Motivation: There are intricate associations between different social and economic factors like inflation, interest rate, consumer price index and housing price index. We can explore how marginal parameters for each of the ''Inflation'' and ''HPI'' distributions, and their correlation, affect their joint and conditional probabilities?
-* ''Caution'': This example is a little different from the [[SOCR_BivariateNormal_JS_Activity#Experiment_1:_Height_vs._Weight|human height and weight experiment above]]. In general, HPI and inflation may not follow normal distributions and may be skewed. Use the [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR Histogram Chart] to plot their distributions. Can the [http://socr.ucla.edu/htmls/HTML5/BivariateNormal/ Bivariate Normal Distribution] be used as an approximate model of the bivariate relation/probabilities of inflation and HPI? How about if we apply a [[SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs|data transformation]]? For example, the figure below shows the result of applying a square-root-transformation to the ''inflation'' variable (\(\lambda=0.5\)). The blue distribution of the transformed data is closer to Normal (note the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments|skewness and kurtosis]]) compared to the red histogram of the raw inflation values.[[Image:SOCR_BivariateNormal_JS_Activity_Fig4.png|300px]]
+* ''Caution'': This example is a little different from the [[SOCR_BivariateNormal_JS_Activity#Experiment_1:_Height_vs._Weight|human height and weight experiment above]]. In general, HPI and inflation may not follow normal distributions and may be skewed. Use the [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR Histogram Chart] to plot their distributions. Can the [http://socr.ucla.edu/htmls/HTML5/BivariateNormal/ Bivariate Normal Distribution] be used as an approximate model of the bivariate relation/probabilities of inflation and HPI? How about if we apply a [[SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs|data transformation]]? For example, the figure below shows the result of applying a square-root-transformation to the ''inflation'' variable (\(\lambda=0.5\)). The blue distribution of the transformed data is closer to Normal (note the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments|skewness and kurtosis]]) compared to the red histogram of the raw inflation values.[[Image:SOCR_BivariateNormal_JS_Activity_Fig6a.png|300px]]
 * Use the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and the [[SOCR_EduMaterials_ModelerActivities_NormalBetaModelFit|SOCR Modeler activity]] to estimate the mean and standard deviation of each of the 2 variables (inflation and HPI).
 * Use the [http://www.socr.ucla.edu/htmls/ana/SimpleRegression_Analysis.html SOCR Simple Linear Regression applet], and the corresponding [[SOCR_EduMaterials_AnalysisActivities_SLR|activity]], to estimate the correlation (\(\rho=Corr(Inflation,HPI)\)).

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 ===Parameter Estimation===
-Use the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] & the [[SOCR_EduMaterials_ModelerActivities_NormalBetaModelFit|modeler activity]] to estimate the mean and standard deviation (SD) of the Height and Weight variables. Also, investigate the distribution of the two variables.
+Use the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] & the [[SOCR_EduMaterials_ModelerActivities_NormalBetaModelFit|modeler activity]] to estimate the mean and standard deviation (SD) of the Height and Weight variables. Also, investigate the distribution of the two variables. By looking at the distributions of the two variables, we can conclude that Height and Weight roughly follow a normal distribution.
-For Height variable:
+* For '''Height''' variable, the distribution is roughly normal.
 <center>[[Image:SOCR_BivariateNormal_JS_Activity_Fig1.png|300px]]</center>
 ==Practice experiments==

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 ==Goals==
 The aims of this activity are to:
-* To clarify the definitions and interplay between marginal, conditional and joint probability distributions (in the bivariate Normal case)
+* Provide a visualization tool for better understanding of the bivariate normal distribution.
-* To learn how to calculate Normal marginal conditional and joint probabilities.
+* Clarify the definitions and interplay between marginal, conditional and joint probability distributions (in the bivariate Normal case).
-* To demonstrate that when X and Y have joint bivariate normal distribution with zero correlation, then X and Y must be independent.
+* To learn how to calculate Normal marginal, conditional and joint probabilities.
 ==Background==
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 ::: \( q(x,y) =  \frac{1}{2} \frac{1}{1-\rho^2} \left ( \left (  \frac{X-\mu_X}{\sigma_X}  \right )^2 -2\rho\frac{X-\mu_X}{\sigma_X}\frac{Y-\mu_Y}{\sigma_Y} +\left (  \frac{Y-\mu_Y}{\sigma_Y}  \right )^2 \right ) \).
-==Requirements==
+==Requirements & usability==
 A modern web-browser with HTML and JavaScript support is required (mobile devices should be fine). The [http://socr.ucla.edu/htmls/HTML5/BivariateNormal/ 3D view of the bivariate Normal distribution] requires [http://www.khronos.org/webgl/ WebGL] support, however this is not absolutely necessary. If you toggle off the "Use WebGL" check-box in the Settings panel you can view the 3D grid/mesh representation of the 2D Normal/Gaussian distribution without WebGL.
 # Go to the [http://socr.ucla.edu/htmls/HTML5/BivariateNormal/ SOCR Bivariate Normal Distribution Webapp].
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 # Probability Results are reported in the bottom text area.
-==Experiment 1: Height vs. Weight==
+==Learning Activity: Human Height and Weight==
 Use the [[SOCR_Data_Dinov_020108_HeightsWeights |SOCR Height vs. Weight dataset]].
 * Motivation: Human heights and weights are correlated, how do the marginal parameters for each of the height and weight distributions, and their correlation, affect the joint and conditional probabilities?
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 ** Joint (e.g., \(P(Height>60 \cap Weight<160)\)).
-==Experiment 2: Inflation vs. HPI==
+===Inflation vs. HPI===
 Use the [[SOCR_Data_MonetaryBaseStocksInterest1959_2009|SOCR Inflation vs. Housing Price Index (HPI) dataset]].
 * Motivation: There are intricate associations between different social and economic factors like inflation, interest rate, consumer price index and housing price index. We can explore how marginal parameters for each of the ''Inflation'' and ''HPI'' distributions, and their correlation, affect their joint and conditional probabilities?
@@ Line 61: / Line 74: @@
 ** Conditional (e.g., \(P(Inflation>5.0 \vert HPI <108)\)),
 ** Joint (e.g., \(P(Inflation>4.0 \cap HPI<110)\)).
 ==References==