SOCR EduMaterials Activities Uniform E EstimateExperiment - Revision history

IvoDinov: /* Description */

2012-11-01T23:31:16Z

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IvoDinov at 19:10, 18 December 2007

2007-12-18T19:10:50Z

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IvoDinov: /* References */

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IvoDinov: /* Description */

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IvoDinov: /* Description */

2007-12-18T17:12:43Z

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IvoDinov at 17:12, 18 December 2007

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IvoDinov at 17:11, 18 December 2007

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 == Description ==
-This experiment is used to estimate the value of the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number e] using simulation. The ''e''-estimate experiment allows us to generate a random sample <math>X_1, X_2, ..., X_n</math> of size n from the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html uniform distribution on (0, 1)]. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded: ''U = minimum n for which the sum'' <math>S = X_1+X_2+...+X_n > 1</math>. That is, <math>U= {\operatorname{argmin}}_n { \left (X_1+X_2+...+X_n > 1 \right )}</math>, note that all <math>X_i \ge 0</math>, so such n exists.
+This experiment is used to estimate the value of the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number e] using simulation. The ''e''-estimate experiment allows us to generate a random sample <math>X_1, X_2, ..., X_n</math> of size n from the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html uniform distribution on (0, 1)]. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistic is recorded: ''U = minimum n for which the sum'' <math>S = X_1+X_2+...+X_n > 1</math>. That is, \( U= {\operatorname{argmin}}_n { \left (X_1+X_2+...+X_n > 1 \right )} \), note that all \( X_i \ge 0 \), so such n exists.
 == Goal ==

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 == Goal ==
-The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea behing a stochastic simulation for estimating the ''natural number e''. If U = minimum n for which the sum <math>S = X_1+X_2+...+X_n > 1</math>, then the expected value of U, <math>E(U)</math>, is approximately equal to the natural number ''e~2.7182...''.
+The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea behind a stochastic simulation for estimating the ''natural number e''. If U = minimum n for which the sum <math>S = X_1+X_2+...+X_n > 1</math>, then the expected value of U, <math>E(U)</math>, is approximately equal to the natural number ''e~2.7182...''.
 == Experiment ==
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 The “update” scroll indicates the number of trials (1, 10, 100, or 1,000) to be performed when selecting the fast forward button and the “stop” scroll indicates the maximum number of trials in the experiment.
-Modifying the sample-size parameter (n) does not change the distribution graph of the events, but will significantly effect the estimates U when rinning the experiment.
+Modifying the sample-size parameter (n) does not change the distribution graph of the events, but will significantly effect the estimates U when running the experiment.
 == Applications ==
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 Estimation of the natural number e is very important in many science and technology developments and studies. There are deterministic algorithms as well as stochastic methods for estimating the value of e. Many of these provide up to 10-billion decimal place accuracy for ''e''.
-This experiment demonstrates an easy to understand, demonstrate and utilize protocol for a stochastic estimation of ''e''. The algorithm may be significantly improved in terms of both speed of convergence and accuracy, relative to sample-size (n). However, the emphasis in this experiment is simplicity and simulation of a trancendental number in real-time using basic tools (sampling from uniform distribution).
+This experiment demonstrates an easy to understand, demonstrate and utilize protocol for a stochastic estimation of ''e''. The algorithm may be significantly improved in terms of both speed of convergence and accuracy, relative to sample-size (n). However, the emphasis in this experiment is simplicity and simulation of a [http://en.wikipedia.org/wiki/Transcendental_number transcendental number] in real-time using basic tools (sampling from uniform distribution).
 ==References==

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 ==See also==
 * [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment | Buffon's Needle experiment (estimating <math>\pi</math> )]]
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 ==References==
 * Russell, K. G.  [http://links.jstor.org/sici?sici=0003-1305%28199102%2945%3A1%3C66%3AETVOEB%3E2.0.CO%3B2-U Estimating the Value of e by Simulation]. The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66-68.
-* Gnedenko, B. V. (1978), The Theory of Probability, Moscow: MIR Publishers, p.194.
+* Gnedenko, B. V. [http://books.google.com/books?id=cLCpDk1XrsIC Theory of Probability], (translated from Russian and published by CRC Press in 1997, original version published in 1982), ISBN 9056995855.
 ==See also==

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 ==References==
-* K. G. Russell. [http://links.jstor.org/sici?sici=0003-1305%28199102%2945%3A1%3C66%3AETVOEB%3E2.0.CO%3B2-U Estimating the Value of e by Simulation]. The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66-68.
+* Russell, K. G.  [http://links.jstor.org/sici?sici=0003-1305%28199102%2945%3A1%3C66%3AETVOEB%3E2.0.CO%3B2-U Estimating the Value of e by Simulation]. The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66-68.
 ==See also==