AP Statistics Curriculum 2007 Distrib RV - Revision history

WikiSysop: /* References */

2014-03-06T17:16:08Z

‎References

IvoDinov at 20:38, 26 April 2013

2013-04-26T20:38:08Z

Jenny: /* Comparing Data and Model Distributions */

2010-06-28T19:36:42Z

‎Comparing Data and Model Distributions

Jenny: /* Examples of Random Variables */

2010-06-28T19:34:12Z

‎Examples of Random Variables

Jenny at 16:58, 28 June 2010

2010-06-28T16:58:10Z

IvoDinov: /* How to Use RVs? */

2010-04-26T15:48:05Z

‎How to Use RVs?

IvoDinov: /* Comparing Data and Model Distributions */

2010-01-27T21:39:56Z

‎Comparing Data and Model Distributions

IvoDinov: /* Comparing Data and Model Distributions */

2010-01-27T21:38:44Z

‎Comparing Data and Model Distributions

IvoDinov: added a Comparing Data and Model Distributions section

2010-01-27T21:35:36Z

added a Comparing Data and Model Distributions section

IvoDinov: added a link to the Problems set

2009-10-26T04:51:52Z

added a link to the Problems set

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 <hr>
-* SOCR Home page: http://www.socr.ucla.edu
+* SOCR Home page: http://www.socr.umich.edu
-{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Distrib_RV}}
+{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_RV}}

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 * '''Card''': Suppose we draw a [[SOCR_EduMaterials_Activities_CardExperiment | 5-card hand from a standard 52-card deck]] and we are interested in the probability that the hand contains at least one pair of cards with identical denomination. Since the sample space of this experiment is large, it should be difficult to list all possible outcomes. However, we can assign a random variable <math>X(s) = \begin{cases}0,& s = \texttt{no-pair},\\
 ,& s = \texttt{at-least-1-pair}.\end{cases}</math> and try to compute the probability of P(X=1), the chance that the hand contains a pair. You can see this explicit RV mapping and the calculations of this probability at the [[SOCR_EduMaterials_Activities_CardExperiment | SOCR Card Experiment]].
 ===Probability density/mass and (cumulative) distribution functions===

@@ Line 67: / Line 67: @@
 ===Comparing Data and Model Distributions===
-To illustrate one example of using distributions for solving practical problems, consider the [[SOCR_Data_Dinov_020108_HeightsWeights | large human weight and height dataset]]. You can use all 25,000 records or just the first 200 of these measurements to follow the protocol below:
+To illustrate one example of using distributions for solving practical problems, we consider the [[SOCR_Data_Dinov_020108_HeightsWeights | large human weight and height dataset]]. You can use all 25,000 records or just the first 200 of these measurements to follow the protocol below:
 * Copy the [[SOCR_Data_Dinov_020108_HeightsWeights| weight and height data]] into the [http://socr.ucla.edu/htmls/SOCR_Charts.html data tab of any of the SOCR Charts] (first clear the default data, then select column 1 heading, and click the '''Paste''' button). This allows you to manipulate each of the 3 data columns independently.
 * Select (highlight with the mouse) one of the columns (e.g., weights or heights) in the SOCR Chart and click the '''Copy''' button. This stores only the data in the chosen column in your mouse buffer.
@@ Line 77: / Line 77: @@
 * You can plug these parameters (mean and standard deviation) into the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Applet] and make inference about your population based on this [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal distribution]] model.
 * Validate that the probabilities of various interesting events (e.g., 68<=Height<70) computed via either using the sample histogram of the data or via the model distribution are very similar.
-* Try fitting another distribution model to your data using the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler]. For example, choose the mixture-of-Normals model ('''MixedFit_Modeler''') and repeat this process? Can you identify possible gender effects in either height or weight of the subjects in your sample? Of so, what are the Male and Female distribution models? Can these be used to predict the gender of subjects (based on their weight or height)?
+* Try fitting another distribution model to your data using the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler]. For example, choose the mixture-of-Normals model ('''MixedFit_Modeler''') and repeat this process. Can you identify possible gender effects in either height or weight of the subjects in your sample? If so, what are the Male and Female distribution models? Can these be used to predict the gender of subjects (based on their weight or height)?
 * Note that the '''Results''' tab also shows some statistics quantifying how good your chosen distribution model is to approximate the (sample) data histogram.

@@ Line 6: / Line 6: @@
 ===Examples of Random Variables===
-* '''Die''': In rolling a regular hexagonal die, the sample space is clearly and numerically well-defined and in this case the random variable is the identity function assigning to each face of the die the numerical value it represents. This the possible outcomes of the RV of this experiment are { 1, 2, 3, 4, 5, 6 }. You can see this explicit RV mapping in the [[SOCR_EduMaterials_Activities_DiceExperiment | SOCR Die Experiment]].
+* '''Die''': In rolling a regular hexagonal die, the sample space is clearly and numerically well-defined. In this case, the random variable is the identity function assigning to each face of the die where the numerical value it represents. This the possible outcomes of the RV of this experiment are { 1, 2, 3, 4, 5, 6 }. You can see this explicit RV mapping in the [[SOCR_EduMaterials_Activities_DiceExperiment | SOCR Die Experiment]].
 * '''Coin''': For a coin toss, a suitable space of possible outcomes is S={H, T} (for heads and tails).  In this case these are not numerical values, so we can define a RV that maps these to numbers. For instance, we can define the RV <math>X: S \longrightarrow [0, 1]</math> as: <math>X(s) = \begin{cases}0,& s = \texttt{H},\\
 ,& s = \texttt{T}.\end{cases}</math>. You can see this explicit RV mapping of heads and tails to numbers in the [[SOCR_EduMaterials_Activities_BinomialCoinExperiment | SOCR Coin Experiment]].
-* '''Card''': Suppose we draw a [[SOCR_EduMaterials_Activities_CardExperiment | 5-card hand from a standard 52-card deck]] and we are interested in the probability that the hand contains at least one pair of cards with identical denomination. Then the sample space of this experiment is large - it should be difficult to list all possible outcomes. However, we can assign a random variable <math>X(s) = \begin{cases}0,& s = \texttt{no-pair},\\
+* '''Card''': Suppose we draw a [[SOCR_EduMaterials_Activities_CardExperiment | 5-card hand from a standard 52-card deck]] and we are interested in the probability that the hand contains at least one pair of cards with identical denomination. Since the sample space of this experiment is large, it should be difficult to list all possible outcomes. However, we can assign a random variable <math>X(s) = \begin{cases}0,& s = \texttt{no-pair},\\
 ,& s = \texttt{at-least-1-pair}.\end{cases}</math> and try to compute the probability of P(X=1), the chance that the hand contains a pair. You can see this explicit RV mapping and the calculations of this probability at the [[SOCR_EduMaterials_Activities_CardExperiment | SOCR Card Experiment]].

@@ Line 2: / Line 2: @@
 === Random Variables===
-A '''random variable''' is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assigns real values to outcomes of experiments. This mapping is called ''random'', as the output values of the mapping depend on the outcome of the experiment, which are indeed random. So, instead of studying the raw outcomes of experiments (e.g., define and compute probabilities), most of the time we study (or compute probabilities) on the corresponding random variables instead. The [http://en.wikipedia.org/wiki/Random_variable formal general definition of random variables may be found here].
+A '''random variable''' is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assigns real values to outcomes of experiments. This mapping is called ''random'', as the output values of the mapping depend on the outcome of the experiment, which are indeed random. So, instead of studying the raw outcomes of experiments (e.g., define and compute probabilities), most of the time we study (or compute probabilities) the corresponding random variables instead. The [http://en.wikipedia.org/wiki/Random_variable formal general definition of random variables may be found here].
 ===Examples of Random Variables===
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 </center>
-The explanation of the [http://en.wikipedia.org/wiki/Benford's_law Benford's Law] may be summarized as follows: The distribution of the first digits must be independent of the measuring units used in observing/recording the integer measurements. For instance, this means that if we had observed length/distance in ''inches'' or ''centimeters'' (inches and centimeters are linearly dependent, <math>1in = 2.54cm</math>), the distribution of the first digit of the measurement must be identical. So, there are about three centimeters for each inch. Thus, the probability that the first digit of a length observation is ''1in'' must be the same as the probability that the first digit of a length in ''centimeters'' starts with either 2 or 3 (with standard round off). Similarly, for observations of ''2in'', need to have their centimeter counterparts either ''5cm'' or ''6cm''. Observations of ''3in'' will correspond to 7 or 8 centimeters, etc. In other words, this distribution must be [http://en.wikipedia.org/wiki/Scale_invariant scale invariant].
+The explanation of the [http://en.wikipedia.org/wiki/Benford's_law Benford's Law] may be summarized as follows: The distribution of the first digits must be independent of the measuring units used in observing/recording the integer measurements. For instance, this means that if we had observed length/distance in ''inches'' or ''centimeters'' (inches and centimeters are linearly dependent, <math>1in = 2.54cm</math>), the distribution of the first digit of the measurement must be identical. So, there are about three centimeters for each inch. Thus, the probability that the first digit of a length observation is ''1in'' must be the same as the probability that the first digit of a length in ''centimeters'' starts with either 2 or 3 (with standard round off). Similarly, for observations of ''2in'', they need to have their centimeter counterparts either ''5cm'' or ''6cm''. Observations of ''3in'' will correspond to 7 or 8 centimeters, etc. In other words, this distribution must be [http://en.wikipedia.org/wiki/Scale_invariant scale invariant].
@@ Line 67: / Line 67: @@
 ===Comparing Data and Model Distributions===
-To illustrate one example of using distributions for solving practical problems consider the [[SOCR_Data_Dinov_020108_HeightsWeights | large human weight and height dataset]]. You can use all 25,000 records or just the first 200 of these measurements to follow the protocol below:
+To illustrate one example of using distributions for solving practical problems, consider the [[SOCR_Data_Dinov_020108_HeightsWeights | large human weight and height dataset]]. You can use all 25,000 records or just the first 200 of these measurements to follow the protocol below:
 * Copy the [[SOCR_Data_Dinov_020108_HeightsWeights| weight and height data]] into the [http://socr.ucla.edu/htmls/SOCR_Charts.html data tab of any of the SOCR Charts] (first clear the default data, then select column 1 heading, and click the '''Paste''' button). This allows you to manipulate each of the 3 data columns independently.
 * Select (highlight with the mouse) one of the columns (e.g., weights or heights) in the SOCR Chart and click the '''Copy''' button. This stores only the data in the chosen column in your mouse buffer.
@@ Line 81: / Line 81: @@
 ===The Web of Distributions===
-There are a large number of families of distributions and distribution classification schemes.
+There is a large number of families of distributions and distribution classification schemes.
 : The most common way to describe the universe of distributions is to partition them into categories. For example, [http://en.wikipedia.org/wiki/Category:Continuous_distributions Continuous Distributions] and [http://en.wikipedia.org/wiki/Category:Discrete_distributions Discrete Distributions]; [http://en.wikipedia.org/wiki/Marginal_distribution marginal] and [http://en.wikipedia.org/wiki/Joint_distribution joint] distributions; [http://en.wikipedia.org/wiki/Probability_distribution#With_finite_support finitely] and [http://en.wikipedia.org/wiki/Probability_distribution#With_infinite_support infinitely] supported, etc.

@@ Line 48: / Line 48: @@
 There are 3 important quantities that we are always interested in when we study random processes. Each of these may be phrased in terms of RVs, which simplifies their calculations.
-* '''Probability Density Function''' (PDF): What is the probability of <math>P(X=x_o)</math>? For instance, in the card example above, we may be interested in [[SOCR_EduMaterials_Activities_CardExperiment#Applications | P(at least 1 pair) = P(X=1) = P(1 pair only) = 0.422569]]. Or in the die example, we may want to know P(Even number turns up) = <math>P(X \in \{2, 4, 6 \}) = 0.5</math>.
+* '''Probability Density Function''' (PDF): What is the probability of <math>P(X=x_o)</math>? For instance, in the card example above, we may be interested in [[SOCR_EduMaterials_Activities_CardExperiment#Applications | P(exactly 1 pair) = P(X=1) = P(1 pair only) = 0.422569]]. Or in the die example, we may want to know P(Even number turns up) = <math>P(X \in \{2, 4, 6 \}) = 0.5</math>.
 * '''Cumulative Distribution Function''' (CDF): <math>P(X <x_o)</math>, for all <math>x_o</math>. For instance, in the (fair) die example we have the following discrete density (mass) and cumulative distribution table:

@@ Line 73: / Line 73: @@
 * Select the ''NormalFit_Modeler'' from the drop-down list on the top-left corner. This is the first model you will be fitting to your data.
 * Select the 3 check-boxes (''Estimate Parameters'', ''Scale Up'', and ''Raw Data'').
-* Go to the '''Graphs''' tab and adjust the 3 slider on the top to get a clear view of your data distribution (sample histogram) and the model distribution function (solid red curve).
+* Go to the '''Graphs''' tab and adjust the 3 sliders on the top to get a clear view of your data distribution (sample histogram) and the model distribution function (solid red curve).
 * The '''Results''' tab will contain the (data-driven) estimates of the parameters for this specific distribution model (in this case [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal]]).
 * You can plug these parameters (mean and standard deviation) into the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Applet] and make inference about your population based on this [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal distribution]] model.
-* Validate that the probabilities of interesting events (e.g., 68<=Height<70) computed via using the sample histogram of the data or via the model distribution are very similar.
+* Validate that the probabilities of various interesting events (e.g., 68<=Height<70) computed via either using the sample histogram of the data or via the model distribution are very similar.
 * Try fitting another distribution model to your data using the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler]. For example, choose the mixture-of-Normals model ('''MixedFit_Modeler''') and repeat this process? can you identify possible gender effects in either height or weight of the subjects in your sample? Of so, what are the Male and Female distribution models? Can these be used to predict the gender of subjects (based on their weight or height)?
 * Note that the '''Results''' tab also shows some statistics quantifying how good your chosen distribution model is to approximate the (sample) data histogram.

← Older revision		Revision as of 21:35, 27 January 2010
Line 65:		Line 65:

	Obviously, we may define a large number of RV for the same process. When are [http://en.wikipedia.org/wiki/Random_variable#Equivalence_of_random_variables two RVs equivalent is dependent on the definition of equivalence]?		Obviously, we may define a large number of RV for the same process. When are [http://en.wikipedia.org/wiki/Random_variable#Equivalence_of_random_variables two RVs equivalent is dependent on the definition of equivalence]?
		+
		+	===Comparing Data and Model Distributions===
		+	To illustrate one example of using distributions for solving practical problems consider the [[SOCR_Data_Dinov_020108_HeightsWeights \| large human weight and height dataset]]. You can use all 25,000 records or just the first 200 of these measurements to follow the protocol below:
		+	* Copy the [[SOCR_Data_Dinov_020108_HeightsWeights\| weight and height data]] into the [http://socr.ucla.edu/htmls/SOCR_Charts.html data tab of any of the SOCR Charts] (first clear the default data, then select column 1 heading, and click the '''Paste''' button). This allows you to manipulate each of the 3 data columns independently.
		+	* Select (highlight with the mouse) one of the columns (e.g., weights or heights) in the SOCR Chart and click the '''Copy''' button. This stores only the data in the chosen column in your mouse buffer.
		+	* Go to the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and paste the data in the first column in the '''Data''' tab using the '''Paste''' button.
		+	* Select the ''NormalFit_Modeler'' from the drop-down list on the top-left corner. This is the first model you will be fitting to your data.
		+	* Select the 3 check-boxes (''Estimate Parameters'', ''Scale Up'', and ''Raw Data'').
		+	* Go to the '''Graphs''' tab and adjust the 3 slider on the top to get a clear view of your data distribution (sample histogram) and the model distribution function (solid red curve).
		+	* The '''Results''' tab will contain the (data-driven) estimates of the parameters for this specific distribution model (in this case [[EBook#Chapter_V:_Normal_Probability_Distribution \|Normal]]).
		+	* You can plug these parameters (mean and standard deviation) into the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Applet] and make inference about your population based on this [[EBook#Chapter_V:_Normal_Probability_Distribution \|Normal distribution]] model.
		+	* Validate that the probabilities of interesting events (e.g., 68<=Height<70) computed via using the sample histogram of the data or via the model distribution are very similar.
		+	* Try fitting another distribution model to your data using the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler]. For example, choose the mixture-of-Normals model ('''MixedFit_Modeler''') and repeat this process? can you identify possible gender effects in either height or weight of the subjects in your sample? Of so, what are the Male and Female distribution models? Can these be used to predict the gender of subjects (based on their weight or height)?
		+	* Note that the '''Results''' tab also shows some statistics quantifying how good your chosen distribution model is to approximate the (sample) data histogram.

	===The Web of Distributions===		===The Web of Distributions===

← Older revision		Revision as of 04:51, 26 October 2009
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	: [http://distributome.org/ The SOCR Distributome applet provides an interactive graphical interface for exploring the relations between different distributions].		: [http://distributome.org/ The SOCR Distributome applet provides an interactive graphical interface for exploring the relations between different distributions].
		+
		+	===[[EBook_Problems_Distrib_RV\|Problems]]===

	<hr>		<hr>