SMHS CLT LLN - Revision history

Dinov: /* References */

2019-08-05T18:53:32Z

‎References

Dinov: /* Software */

2015-07-29T19:09:34Z

‎Software

Jslavine: /* Law of Large NUmbers (LLN) */

2014-10-01T15:11:30Z

‎Law of Large NUmbers (LLN)

Dinov: /* Central Limit Theorem (CLT) */

2014-09-29T22:46:00Z

‎Central Limit Theorem (CLT)

Dinov: /* Central Limit Theorem (CLT) */

2014-09-29T13:17:36Z

‎Central Limit Theorem (CLT)

Dinov: /* Central Limit Theorem (CLT) */

2014-09-29T13:16:08Z

‎Central Limit Theorem (CLT)

Dinov: /* Central Limit Theorem (CLT) */

2014-09-29T13:15:29Z

‎Central Limit Theorem (CLT)

Dinov: /* Central Limit Theorem (CLT) */

2014-09-16T19:46:26Z

‎Central Limit Theorem (CLT)

Dinov: /* Problems */

2014-09-15T17:11:50Z

‎Problems

Dinov: /* Problems */

2014-09-15T17:08:46Z

‎Problems

@@ Line 198: / Line 198: @@
 ===References===
-http://www.amstat.org/publications/jse/v16n2/dinov.html
+* [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3095954/ Law of Large Numbers: the Theory, Applications and Technology-based Education]
 <hr>

@@ Line 133: / Line 133: @@
 ===Software===
 * [http://socr.ucla.edu/htmls/exp/LLN_Simple_Experiment.html SOCR LLN Experiment (Java Applet)]
-* [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html SOCR COin Toss LLN Experiment (Java Applet)]
+* [http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html SOCR Coin Toss LLN Experiment (Java Applet)]
 * [[SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_.CF.80_using_SOCR_simulation |SOCR LLN Activity]]
 * [[SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem| SOCR CLT Activity]]

@@ Line 8: / Line 8: @@
 ===Theory===
-====Law of Large NUmbers (LLN)====
+====Law of Large Numbers (LLN)====
 When performing the same experiment a large number of times, the average of the results obtained should be close to the expected value and tends to get closer to the expected value with increasing number of trials.
 *It is generally necessary to draw the parallels between the formal LLN statements (in terms of sample averages) and the frequent interpretations of the LLN (in terms of probabilities of various events). Suppose we observe the same process independently multiple times. Assume a binarized (dichotomous) function of the outcome of each trial is of interest (e.g., failure may denote the event that the continuous voltage measure $< 0.5V$, and the complement, success, that voltage $≥ 0.5V,$ this is the situation in electronic chips which binarize electric currents to 0 or 1). Researchers are often interested in the event of observing a success at a given trial or the number of successes in an experiment consisting of multiple trials. Let’s denote $p=P(success)$ at each trial. Then, the ratio of the total number of successes to the number of trials $(n)$ is the average:$\bar X_{n}=\frac{1}{n}\sum_{i=1}^{n}X_{i}$ , where $X_{i}=0$ if failure and 1 if success. Hence, $\bar X_{n}=\hat\rho$,the ratio of the observed frequency of that event to the total number of repetitions, estimates the true p=P(success). Therefore, $\hat\rho$ converges towards $\rho$ as the number of (identical and independent) trials increases.

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 ====Central Limit Theorem (CLT)====
 The arithmetic mean of a sufficiently large number of iterates of independent random variables given certain conditions will be approximately normally distributed. It states that the sum of many independent and identically distributed (i.i.d.) random variables will tend to be distributed according to one of a small set of attractor distributions. There are various statements of the central limit theorem, but all of them represent weak-convergence results regarding (mostly) the sums of independent identically-distributed (random) variables.
-*Definition of CLT: let ${X_{1},X_{2},…,X_{n}}$ be a i.i.d. random sample of size n drawn from distributions of expected values $\mu$ and finite variance $\sigma^{2}$. The sample average $\bar{x}_{n}=\frac{X_{1}+X_{2}+⋯+X_{n}}{n}$. By LLT, the sample averages converge in probability and almost surely to the expected value $\mu$ as $n\rightarrow \infty$. As n gets larger, the distribution of difference between the sample average $\bar{x}_{n}$ and its limit $\mu$, when multiplied by the factor $\sqrt n$, that is $\sqrt n(\bar{x}_{n}-\mu)$ approximates the normal distribution with mean 0 and variance $\sigma^{2}$: $\sqrt n(\bar{x}_{n}-\mu)\rightarrow N(0,\sigma^{2})$ when n is large enough. Thus, for large enough n, the distribution of $\bar{x}_{n}$ is close to the normal distribution with mean $\mu$ and variance $\frac{\sigma^{2}}{n}$: $S_{n}\rightarrow N(\mu,\frac{\sigma^{2}}{n})$.
+*Definition of CLT: let ${X_{1},X_{2},…,X_{n}}$ be a i.i.d. random sample of size n drawn from distributions of expected values $\mu$ and finite variance $\sigma^{2}$. The sample average $\bar{x}_{n}=\frac{X_{1}+X_{2}+⋯+X_{n}}{n}$. By LLT, the sample averages converge in probability and almost surely to the expected value $\mu$ as $n\rightarrow \infty$. As n gets larger, the distribution of difference between the sample average $\bar{x}_{n}$ and its limit $\mu$, when multiplied by the factor $\sqrt n$, that is $\sqrt n(\bar{x}_{n}-\mu)$ approximates the normal distribution with mean 0 and variance $\sigma^{2}$: $\sqrt n(\bar{x}_{n}-\mu)\rightarrow N(0,\sigma^{2})$ when n is large enough. Thus, for large enough n, the distribution of $\bar{x}_{n}$ is close to the normal distribution with mean $\mu$ and variance $\frac{\sigma^{2}}{n}$: $\bar{x}_{n}\rightarrow N(\mu,\frac{\sigma^{2}}{n})$.
 * See the [[SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem| Generalized CLT Activity and Applications]].

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 The arithmetic mean of a sufficiently large number of iterates of independent random variables given certain conditions will be approximately normally distributed. It states that the sum of many independent and identically distributed (i.i.d.) random variables will tend to be distributed according to one of a small set of attractor distributions. There are various statements of the central limit theorem, but all of them represent weak-convergence results regarding (mostly) the sums of independent identically-distributed (random) variables.
 *Definition of CLT: let ${X_{1},X_{2},…,X_{n}}$ be a i.i.d. random sample of size n drawn from distributions of expected values $\mu$ and finite variance $\sigma^{2}$. The sample average $\bar{x}_{n}=\frac{X_{1}+X_{2}+⋯+X_{n}}{n}$. By LLT, the sample averages converge in probability and almost surely to the expected value $\mu$ as $n\rightarrow \infty$. As n gets larger, the distribution of difference between the sample average $\bar{x}_{n}$ and its limit $\mu$, when multiplied by the factor $\sqrt n$, that is $\sqrt n(\bar{x}_{n}-\mu)$ approximates the normal distribution with mean 0 and variance $\sigma^{2}$: $\sqrt n(\bar{x}_{n}-\mu)\rightarrow N(0,\sigma^{2})$ when n is large enough. Thus, for large enough n, the distribution of $\bar{x}_{n}$ is close to the normal distribution with mean $\mu$ and variance $\frac{\sigma^{2}}{n}$: $S_{n}\rightarrow N(\mu,\frac{\sigma^{2}}{n})$.
 *Multidimensional CLT: extend the central limit theorem characteristics to the cases where $X_1,X_2,…,X_n$ for each individual is an i.i.d. random vector in $R^k$ with mean $μ=E(X_i)$ and covariance matrix $Σ$. Then with multidimensional CLT, Let ${X_i}=\begin{bmatrix} X_{i(1)} \\ \vdots \\ X_{i(k)} \end{bmatrix}$ be the $i$-vector. The bold in '''X'''<sub>''i''</sub> means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be $\begin{bmatrix} X_{1(1)} \\ \vdots \\ X_{1(k)} \end{bmatrix}+\begin{bmatrix} X_{2(1)} \\ \vdots \\ X_{2(k)} \end{bmatrix}+\cdots+\begin{bmatrix} X_{n(1)} \\ \vdots \\ X_{n(k)} \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^{n} \left [ X_{i(1)} \right ] \\ \vdots \\ \sum_{i=1}^{n} \left [ X_{i(k)} \right ] \end{bmatrix} = \sum_{i=1}^{n} \left [ \mathbf{X_i} \right ].$ Also, the average will be $\left (\frac{1}{n}\right)\sum_{i=1}^{n} \left [ \mathbf{X_i} \right ]= \frac{1}{n}\begin{bmatrix} \sum_{i=1}^{n} \left [ X_{i(1)} \right ] \\ \vdots \\ \sum_{i=1}^{n} \left [ X_{i(k)} \right ] \end{bmatrix} = \begin{bmatrix} \bar X_{i(1)} \\ \vdots \\ \bar X_{i(k)} \end{bmatrix}=\mathbf{\bar X_n}$. Thus, $\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left [\mathbf{X_i} - E\left ( X_i\right ) \right ]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \left [ \mathbf{X_i} - \mu \right ]=\sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right) .$

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 How would you go about in practice if you had to estimate $e^2≈7.38861124$?
-Similarly, try to estimate π≈3.141592 and π^2≈9.8696044 using the SOCR Buffon’s Needle Experiment.
+Similarly, try to estimate $π≈3.141592$ and $π^2≈9.8696044$ using the [[SOCR_EduMaterials_Activities_BuffonNeedleExperiment|SOCR Buffon’s Needle Experiment]].
-.5) Run the SOCR Roulette Experiment and bet on 1-18 (out of the 38 possible numbers/outcomes).
+.5) Run the [[SOCR_EduMaterials_Activities_RouletteExperiment|SOCR Roulette Experiment]] and bet on 1-18 (out of the 38 possible numbers/outcomes). What is the probability of success (p)?
-What does the LLN imply about p and repeated runs of this experiment?
+What does the LLN imply about $p$ and repeated runs of this experiment?
 Run this experiment 3 times. What is the sample estimate of p ($\hat{p}$)? What is the difference $p-\hat{p}$?
@@ Line 194: / Line 193: @@
 When can we reasonably expect statistics, other than the sample mean, to have CLT properties?
-If a native process has $σ_X  = 10$ and we take a sample of size 10, what will be (σ_{\bar{X}})? Does it depend on the shape of the original process? How large should the sample-size be so that $σ_{\bar{X}}=2/3 σ_X$?
+If a native process has $σ_X  = 10$ and we take a sample of size 10, what will be ($σ_{\bar{X}}$)? Does it depend on the shape of the original process? How large should the sample-size be so that $σ_{\bar{X}}=\frac{2}{3} σ_X$?
 ===References===

@@ Line 163: / Line 163: @@
-.4) Use the SOCR Uniform e-Estimate Experiment to obtain stochastic estimates of the natural number e≈2.7182.
+.4) Use the SOCR Uniform e-Estimate Experiment to obtain stochastic estimates of the natural number $e≈2.7182$.
-Try to explain in words, and support your argument with data/results from this simulation, why is the expected value of the variable U (defined above) equal to e,E(U) = e.
+Try to explain in words, and support your argument with data/results from this simulation, why is the expected value of the variable U (defined above) equal to e, $E(U) = e$.
 How does the LLN come into play in this experiment?
-How would you go about in practice if you had to estimate e^2≈7.38861124?
+How would you go about in practice if you had to estimate $e^2≈7.38861124$?
 Similarly, try to estimate π≈3.141592 and π^2≈9.8696044 using the SOCR Buffon’s Needle Experiment.
@@ Line 179: / Line 179: @@
 What does the LLN imply about p and repeated runs of this experiment?
-Run this experiment 3 times. What is the sample estimate of p (p ̂)? What is the difference p-p ̂?
+Run this experiment 3 times. What is the sample estimate of p ($\hat{p}$)? What is the difference $p-\hat{p}$?
 Would this difference change if we ran the experiment 10 or 100 times? How?
@@ Line 194: / Line 194: @@
 When can we reasonably expect statistics, other than the sample mean, to have CLT properties?
-If a native process has σ_X  = 10 and we take a sample of size 10, what will be (σ_X ) ̅? Does it depend on the shape of the original process? How large should the sample-size be so that (σ_X ) ̅=2/3 σ_X?
+If a native process has $σ_X  = 10$ and we take a sample of size 10, what will be (σ_{\bar{X}})? Does it depend on the shape of the original process? How large should the sample-size be so that $σ_{\bar{X}}=2/3 σ_X$?
 ===References===