SOCR EduMaterials Activities LawOfLargeNumbers - Revision history

IvoDinov at 22:33, 16 January 2009

2009-01-16T22:33:36Z

IvoDinov: /* Overview */

2009-01-16T18:23:35Z

‎Overview

IvoDinov: /* Experiment 3= */

2008-04-28T04:20:07Z

‎Experiment 3=

IvoDinov: added the multi-nomial experiment demonstrating LLN

2008-04-28T04:19:28Z

added the multi-nomial experiment demonstrating LLN

IvoDinov: Added hands-on activities

2008-01-11T03:16:12Z

Added hands-on activities

IvoDinov: /* Estimating ''e'' using SOCR simulation */

2008-01-10T22:38:11Z

‎Estimating ''e'' using SOCR simulation

IvoDinov at 19:19, 18 December 2007

2007-12-18T19:19:18Z

IvoDinov at 19:18, 18 December 2007

2007-12-18T19:18:36Z

IvoDinov: /* '''Exercise 1''' */

2007-12-18T19:17:31Z

‎'''Exercise 1'''

IvoDinov at 19:16, 18 December 2007

2007-12-18T19:16:54Z

@@ Line 106: / Line 106: @@
 * '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).
 * '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!
 <hr>

@@ Line 2: / Line 2: @@
 == Overview==
-This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.
+This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments. The [http://socr.ucla.edu/htmls/exp/LLN_Simple_Experiment.html SOCR LLN applet is available here].
 ===Goals of the SOCR LLN activity===

@@ Line 51: / Line 51: @@
 Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping  needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <math>{ 2 \over \pi} \approx 0.63662</math>. In practice, to estimate <math>\pi</math> from a number of needle drops (N), we take the reciprocal of the sample odds-of-intersection.
-==Experiment 3===
+==Experiment 3==
 Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).
 <center>

← Older revision		Revision as of 04:19, 28 April 2008
Line 50:		Line 50:

	Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 \| SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <math>{ 2 \over \pi} \approx 0.63662</math>. In practice, to estimate <math>\pi</math> from a number of needle drops (N), we take the reciprocal of the sample odds-of-intersection.		Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 \| SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <math>{ 2 \over \pi} \approx 0.63662</math>. In practice, to estimate <math>\pi</math> from a number of needle drops (N), we take the reciprocal of the sample odds-of-intersection.
		+
		+	==Experiment 3===
		+	Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={3 ones, 3 twos, 2 threes, and 2 fours}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).
		+	<center>
		+	{\| class="wikitable" style="text-align:center; width:75%" border="1"
		+	\|-
		+	\| ''x'' \|\| 1 \|\| 2 \|\| 3 \|\| 4 \|\| 5 \|\| 6
		+	\|-
		+	\| ''P(X=x)'' \|\| 0.286 \|\| 0.238 \|\| 0.19 \|\| 0.143 \|\| 0.095 \|\| 0.048
		+	\|}
		+	</center>
		+
		+	: ''P(A)=?''
		+
		+	Of course, we can compute this number exactly as:
		+
		+	: <math>P(A) = {10! \over 3!\times 3! \times 2! \times 2! } \times 0.286^3 \times 0.238^3\times 0.19^2 \times 0.143^2 = 0.00586690138260962656816896.</math>
		+
		+	However, we can also find a pretty close empirically-driven estimate using the [[SOCR_EduMaterials_Activities_DiceExperiment \| SOCR Dice Experiment]].
		+
		+	For instance, running the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Dice Experiment] 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.
		+
		+	<center>[[Image:SOCR_EBook_Dinov_Multinimial_030508_Fig1.jpg\|500px]]</center>
		+
		+	Now, we can actually count how many of these 1,000 trials generated the event ''A'' as an outcome. In one such experiment of 1,000 trials, there were 8 outcomes of the type {3 ones, 3 twos, 2 threes and 2 fours}. Therefore, the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above
		+	: <math>P(A) \approx {8 \over 1,000}=0.008</math>.
		+
		+	Note that that this approximation is close to the exact answer above. By the Law of Large Numbers, we know that this SOCR empirical approximation to the exact multinomial probability of interest will significantly improve as we increase the number of trials in this experiment to 10,000.

	== Hands-on activities==		== Hands-on activities==

@@ Line 3: / Line 3: @@
 == Overview==
 This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.
 ===Example===
@@ Line 14: / Line 20: @@
 == Exercise 1==
-This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment consist of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>.
+This exercise illustrates the statement and validity of the LLN in the situation of tossing (biased or fair) coins repeatedly. Suppose we let H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are <math>0<p<1</math> and <math>0<1-p<1</math>, respectfully. The sample space of this experiment consists of sequences of H's and Ts. For example, an outcome may be <math>\{H, H, T, H, H, T, T, T, ....\}</math>. If we toss a coin n times, the size of the sample-space is <math>2^n</math>, as the coin tosses are independent. [[About_pages_for_SOCR_Distributions | Binomial Distribution]] governs the probability of observing <math>0\le k\le n</math> Heads in <math>n</math> experiments, which is evaluated by the binomial density at <math>k</math>.
 In this case we will be interested in two random variables associated with this process. The first variable will be the ''proportion of Heads'' and the second will be the ''differences of the number of Heads and Tails''. This will empirically demonstrate the LLN and its most common misconceptions (presented below). Point your browser to the [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. From this table, you can copy and paste the summary for further processing using other computational resources (e.g., [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] or [http://office.microsoft.com/excel MS Excel]).
@@ Line 44: / Line 50: @@
 Similarly, one may approximate the transcendental number <math>\pi</math>, using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 | SOCR Buffon’s Needle Experiment]]. Here, the LLN again provides the foundation for a better approximation of <math>\pi</math> by virtually dropping  needles (many times) on a tiled surface and observing if the needle crosses a tile grid-line. For a tile grid of size 1, the odds of a needle-line intersection are <math>{ 2 \over \pi} \approx 0.63662</math>. In practice, to estimate <math>\pi</math> from a number of needle drops (N), we take the reciprocal of the sample odds-of-intersection.
 == Other SOCR LLN Activities==

@@ Line 37: / Line 37: @@
 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://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''.
 <center>[[Image:SOCR_Activities_Uniform_E_EstimateExperiment_Dinov_121907_Fig1.jpg|400px]]</center>

@@ Line 4: / Line 4: @@
 This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II]] and [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III]] of this activity contain more examples and diverse experiments.
-==== Example ====
+===Example===
 The average weight of 10 students from a class of 100 students is most likely closer to the ''real average'' weight of all 100 students, compared to the average weight of 3 randomly chosen students from that same class. This is because the sample of 10 is a ''larger number'' than the sample of only 3 and better represents the entire class. At the extreme, a sample of 99 of the 100 students will produce a sample average almost exactly the same as the average for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.
-==== Statement of the Law of Large Numbers ====
+===Statement of the Law of Large Numbers===
 If an event of probability p is observed repeatedly during '''independent repetitions''', the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.
-==== The theory behind the LLN ====
+===The theory behind the LLN===
 Complete details about the ''weak'' and ''strong'' laws of large numbers may be found [http://en.wikipedia.org/wiki/Law_of_large_numbers here].
@@ Line 28: / Line 28: @@
 * Remember that the more experiments you run the closer the theoretical and sample proportions will be (by LLN). Go in '''Continuous run mode''' and watch the convergence of the sample proportion to <math>p</math>. Can you explain in words, why can't we expect the second variable of interest (the differences of Heads and Tails) to converge? [[Image:SOCR_Activities_LLN_Dinov_022007_Fig2.jpg|200px]]
-== '''Exercise 2'''==
+==Exercise 2==
 The second SOCR demonstration of the law of large numbers will be quite different and practically useful. Here we show how the LLN implies 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''].
@@ Line 49: / Line 49: @@
 * [[SOCR_EduMaterials_Activities_LawOfLargeNumbersExperiment | Part III of this activity]]
-== '''Common Misconceptions regarding the LLN'''==
+== Common Misconceptions regarding the LLN==
 * '''Misconception 1''': If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the <math>11^{th}</math> trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a ''memoryless'' process).
 * '''Misconception 2''': If run large number of coin tosses, the '''number of heads''' and '''number of tails''' become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!

@@ Line 1: / Line 1: @@
 == [[SOCR_EduMaterials_Activities | SOCR Educational Materials - Activities]] - SOCR Law of Large Numbers Activity ==
-== This is part I of a heterogeneous activity that demonstrates the theory and applications of the Law of Large Numbers (LLN). [[SOCR_EduMaterials_Activities_LawOfLargeNumbers2 | Part II of this activity]] contains more examples and diverse experiments. ==
+== Overview==
 ==== Example ====