AP Statistics Curriculum 2007 Limits LLN - Revision history

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

2020-03-03T16:44:21Z

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

IvoDinov: /* LLN Application */

2012-02-17T16:49:12Z

‎LLN Application

IvoDinov: /* SOCR LLN Activity */

2012-02-17T16:48:14Z

‎SOCR LLN Activity

IvoDinov: /* See also */ typo

2011-05-09T17:02:01Z

‎See also: typo

IvoDinov: added see also section

2011-05-09T17:01:46Z

added see also section

Jenny: /* Motivation */

2010-06-29T17:44:38Z

‎Motivation

Jenny: /* Motivation */

2010-06-28T20:19:01Z

‎Motivation

IvoDinov: /* SOCR LLN Activity */ typo

2010-05-24T15:28:05Z

‎SOCR LLN Activity: typo

IvoDinov: added a link to the Problems set

2009-10-26T05:00:46Z

added a link to the Problems set

IvoDinov: /* LLN Application */

2008-11-05T20:12:32Z

‎LLN Application

← Older revision		Revision as of 16:44, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Limits_LLN}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Limits_LLN}}

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 Now, the expected value <math>E(U) = e \approx 2.7182</math>. Therefore, by LLN, taking averages of <math>\left \{ U_1, U_2, U_3, ..., U_k \right \}</math> values, each computed from random samples <math>X_1, X_2, ..., X_n \sim U(0,1)</math> as described above, will provide a more accurate estimate (as <math>k \rightarrow \infty</math>) of the natural number ''e''.
-The '''Uniform E-Estimate Experiment''', part of [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments], provides a hands-on demonstration of how the LLN facilitates stochastic simulation-based estimation of ''e''.
+The '''Uniform E-Estimate Experiment''', part of [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html SOCR Experiments], provides a hands-on demonstration of how the LLN facilitates stochastic simulation-based estimation of ''e''.
 <center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg|400px]]</center>

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 The [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | complete SOCR LLN Activity is available here]].
-Go to [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the '''Coin Toss LLN Experiment''' from the drop-down list of experiments in the top-left panel. This applet consists of a control toolbar on the top followed by a graph panel in the middle and a results table at the bottom. Use the toolbar to flip coins one at a time, 10, 100, 1,000 at a time or continuously! The toolbar also allows you to stop or reset an experiment and select the probability of Heads ('''p''') using the slider. The graph panel in the middle will dynamically plot the values of the two variables of interest (''proportion of heads'' and ''difference of Heads and Tails''). The outcome table at the bottom presents the summaries of all trials of this experiment.
+Go to [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html SOCR Experiments] and select the '''Coin Toss LLN Experiment''' from the drop-down list of experiments in the top-left panel. This applet consists of a control toolbar on the top followed by a graph panel in the middle and a results table at the bottom. Use the toolbar to flip coins one at a time, 10, 100, 1,000 at a time or continuously! The toolbar also allows you to stop or reset an experiment and select the probability of Heads ('''p''') using the slider. The graph panel in the middle will dynamically plot the values of the two variables of interest (''proportion of heads'' and ''difference of Heads and Tails''). The outcome table at the bottom presents the summaries of all trials of this experiment.
 ===LLN Application===

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 ===See also===
 * [[SOCR_EduMaterials_Activities_LawOfLargeNumbers| The SOCR Law of Large Numbers Activity]]
-* [[SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_.CF.80_using_SOCR_simulation| Estimating π using SOCR simulation]
+* [[SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_.CF.80_using_SOCR_simulation| Estimating π using SOCR simulation]]
 * [http://socr.ucla.edu/htmls/exp/LLN_Simple_Experiment.html Simple LLN Experiment Applet]
 * [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html Coin Toss LLN Experiment Applet]

← Older revision		Revision as of 17:01, 9 May 2011
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	<center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg\|400px]]</center>		<center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg\|400px]]</center>
		+
		+	===See also===
		+	* [[SOCR_EduMaterials_Activities_LawOfLargeNumbers\| The SOCR Law of Large Numbers Activity]]
		+	* [[SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_.CF.80_using_SOCR_simulation\| Estimating π using SOCR simulation]
		+	* [http://socr.ucla.edu/htmls/exp/LLN_Simple_Experiment.html Simple LLN Experiment Applet]
		+	* [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html Coin Toss LLN Experiment Applet]

	<hr>		<hr>

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 === Motivation===
-Suppose we ''independently'' conduct one experiment repeatedly. Assume that we are interested in the relative frequency of occurrence of one event whose probability to be observed at each experiment is ''p''. The ratio of the observed frequency of that event to the total number of repetitions converges towards ''p'' as the number of (identical and independent) experiments increases. This is an informal statement of the Law of Large Numbers (LLN).
+Suppose we ''independently'' conduct one experiment repeatedly. Assume that we are interested in the relative frequency of the occurrence of one event whose probability to be observed at each experiment is ''p''. The ratio of the observed frequency of that event to the total number of repetitions converges towards ''p'' as the number of (identical and independent) experiments increases. This is an informal statement of the Law of Large Numbers (LLN).
 For a more concrete example, suppose we study the average height of a class of 100 students. Compared to the average height of 3 randomly chosen students from this class, the average height of 10 randomly chosen students is most likely closer to the real average height of all 100 students. Since the sample of 10 is a larger than the sample of 3, it is a better representation of the entire class. At one extreme, a sample of 99 of the 100 students will produce a sample average height almost exactly the same as the average height for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.

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 Suppose we ''independently'' conduct one experiment repeatedly. Assume that we are interested in the relative frequency of occurrence of one event whose probability to be observed at each experiment is ''p''. The ratio of the observed frequency of that event to the total number of repetitions converges towards ''p'' as the number of (identical and independent) experiments increases. This is an informal statement of the Law of Large Numbers (LLN).
-For a more concrete example, suppose we study the average height of a class of 100 students. Compared to the average height of 3 randomly chosen students from this class, the average height of 10 randomly chosen students is most likely closer to the real average height of all 100 students. Because the sample of 10 is a larger than the sample of 3, it is a better representation of the entire class. At one extreme, a sample of 99 of the 100 students will produce a sample average height almost exactly the same as the average height for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.
+For a more concrete example, suppose we study the average height of a class of 100 students. Compared to the average height of 3 randomly chosen students from this class, the average height of 10 randomly chosen students is most likely closer to the real average height of all 100 students. Since the sample of 10 is a larger than the sample of 3, it is a better representation of the entire class. At one extreme, a sample of 99 of the 100 students will produce a sample average height almost exactly the same as the average height for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.
 ===The Law of Large Numbers (LLN)===

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 ===SOCR LLN Activity===
-The [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | complete SOCR LLN Activity is availabel here]].
+The [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | complete SOCR LLN Activity is available here]].
 Go to [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the '''Coin Toss LLN Experiment''' from the drop-down list of experiments in the top-left panel. This applet consists of a control toolbar on the top followed by a graph panel in the middle and a results table at the bottom. Use the toolbar to flip coins one at a time, 10, 100, 1,000 at a time or continuously! The toolbar also allows you to stop or reset an experiment and select the probability of Heads ('''p''') using the slider. The graph panel in the middle will dynamically plot the values of the two variables of interest (''proportion of heads'' and ''difference of Heads and Tails''). The outcome table at the bottom presents the summaries of all trials of this experiment.

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 <center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg|400px]]</center>
 <hr>
 ===References===
-* TBD
+* Dinov, ID, Christou, N, and Gould, R (2009) [http://www.amstat.org/publications/jse/v17n1/dinov.html Law of Large Numbers: the Theory, Applications and Technology-based Education]. [http://www.amstat.org/publications/jse JSE, Vol. 17, No. 1], 1-15, 2009.
 <hr>

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 One demonstration of the law of large numbers provides practical algorithms for estimation of [http://en.wikipedia.org/wiki/Transcendental_number transcendental numbers]. The two most popular transcendental numbers are [http://en.wikipedia.org/wiki/Pi <math>\pi</math>] and [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 ''e''].
-The [[SOCR_EduMaterials_Activities_Uniform_E_EstimateExperiment | SOCR E-Estimate Experiment]] provides the complete details of this simulation. In a nutshell, we can estimate the value of the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number e] using random sampling from Uniform distribution. Suppose <math>X_1, X_2, \cdots, X_n</math> are drawn from [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html uniform distribution on (0, 1)] and define <math>U= {\operatorname{argmin}}_n { \left (X_1+X_2+...+X_n > 1 \right )}</math>, note that all <math>X_i \ge 0</math>.
+The [[SOCR_EduMaterials_Activities_Uniform_E_EstimateExperiment | SOCR E-Estimate Experiment]] provides the complete details of this simulation. In a nutshell, we can estimate the value of the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number e] using random sampling from [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Uniform distribution]. Suppose <math>X_1, X_2, \cdots, X_n</math> are drawn from [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Uniform distribution on (0, 1)] and define <math>U= {\operatorname{argmin}}_n { \left (X_1+X_2+...+X_n > 1 \right )}</math>, note that all <math>X_i \ge 0</math>.
 Now, the expected value <math>E(U) = e \approx 2.7182</math>. Therefore, by LLN, taking averages of <math>\left \{ U_1, U_2, U_3, ..., U_k \right \}</math> values, each computed from random samples <math>X_1, X_2, ..., X_n \sim U(0,1)</math> as described above, will provide a more accurate estimate (as <math>k \rightarrow \infty</math>) of the natural number ''e''.
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 <hr>
 ===References===
 * TBD