Difference between revisions of "SOCR EduMaterials Activities Uniform E EstimateExperiment"
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== Description == | == 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 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, | + | 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 == | == Goal == | ||
| − | The purpose of the Uniform E-Estimate Experiment is to provide an interactive computer demonstration illustrating a simple idea | + | 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 == | == 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. | 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 | + | 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 == | == 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''. | 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 | + | 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== | ==References== | ||
| − | * K. G. | + | * 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. [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== | ==See also== | ||
| − | * [[ | + | * [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment | Buffon's Needle experiment (estimating <math>\pi</math> )]] |
| − | + | * [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers Activity]] | |
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Latest revision as of 18:31, 1 November 2012
Contents
SOCR Experiments - Uniform e-Estimate Experiment
Description
This experiment is used to estimate the value of the natural number e using simulation. The e-estimate experiment allows us to generate a random sample \(X_1, X_2, ..., X_n\) of size n from the 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 \(S = X_1+X_2+...+X_n > 1\). 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
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 \(S = X_1+X_2+...+X_n > 1\), then the expected value of U, \(E(U)\), is approximately equal to the natural number e~2.7182....
Experiment
Go to the SOCR Experiments and select the Uniform E-Estimate Experiment from the drop-down list of experiments on the top left. The image below shows the initial view of this experiment:
When pressing the play button, one trial will be executed and recorded in the distribution table below. The fast forward button symbolizes the n-th number of trials to be executed each time. The stop button ceases any activity and is helpful when the experimenter chooses “continuous run.” The fourth button will reset the entire experiment, deleting all previous information and data collected.
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 running the experiment.
Applications
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 transcendental number in real-time using basic tools (sampling from uniform distribution).
References
- Russell, K. G. Estimating the Value of e by Simulation. The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66-68.
- Gnedenko, B. V. 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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