"Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed" ([Wikipedia](http://en.wikipedia.org/wiki/Central_limit_theorem)).

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Using central limit theorem for approximation

Let $X$ be a random varaible from a distribution with pdf $$ f(x) = \theta x^{\theta-1}, \quad 0< x < 1. $$ a) Name the distribution of $U=-\ln(X)$ by first finding its density ...
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59 views

Does heteroskedasticity matter if you have a large enough sample?

Let's say you run a regression with over 200 observations. Would this reasonably large sample mitigate the impact of residuals heteroskedasticity as an offshoot of the Central Limit Theorem, or ...
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1answer
21 views

Can I pull a confidence interval out of a single sample by dividing it into sub-samples?

Let's say I have taken a sample $S$ of a population. I am trying to figure out the population mean. Because I have only made one sample, the best I can do is assume that $\bar S$ is the mean of the ...
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1answer
45 views

Does the Central Limit Theorem only work for iid random variables?

Can we say anything about the distribution of the sum of not iid random variables?
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150 views

Convergence in Distribution\CLT

Given that $N = n$, the conditional distr. of $Y$ is $\chi ^2(2n)$. $N$ has marginal distr. of Poisson($\theta$), $\theta$ is a positive constant. Show that, as $\theta \rightarrow \infty$, $\space ...
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58 views

Example of CLT when moments do not exist

Consider $X_n = \begin{cases} 1 & w.p (1 - 2^{-n})/2\\ -1 &w.p~ (1 - 2^{-n})/2\\ 2^{k} &w.p~ 2^{-k} \text{ for } k > n\\ \end{cases}$ I need to show that even though this has ...
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1answer
38 views

CLT can be used for weighted sum of different Bernoulli variables?

Suppose $$ z_i \sim Bernoulli (p_i) $$ Can we use CLT for the following weighted sum? $$ S = \sum_i w_i z_i $$ i.e. can $S$ be approximated with a normal distribution? If yes, with which theorem? ...
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88 views

central limit theorem

can I do my statistics work based on the central limit theorem? I need to perform a t-test, ANOVA and multiple regression. my outcome variable is highly not normally distributed (Highly positively ...
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48 views

Self-study question

I'm currently working on a self study worksheet. I understand most parts of the solution for part III, but I can't seem to make out how this comes about: QUESTION: ANSWER:
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30 views

Central limit theorem with scilab

I'm trying to illustrate the CLT with scilab but my results are weird. Did I make a mistake ? ...
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34 views

Shouldn't a function of data from a PDF repeated over and over on new data eventually yield a Gaussian PDF?

I got into an interesting discussion with a co-worker today and we are not sure what the answer is: We have $N=1000$ samples from a Rayleigh PDF. We take those $N$ samples, and compute their ...
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31 views

Central Limit Theorem: Likelihood multiplication

Been watching this video by Tom Minka on Expectation Propagation (http://videolectures.net/mlss09uk_minka_ai/). At about 19:12, he says that the reason the moment matching technique works is that when ...
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126 views

Can a normal approximation assumption justify itself?

I am learning elementary statistics. I found an exercise, which asks to compute the desired sample size for some interval for standard error. The solution, in class slides, first assumes the sample ...
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88 views

Convergence in distribution (central limit theorem)

If $X_1, ... , X_n$ be iid exponential with mean $1/\lambda$. Let $S_n = X_1 + ... + X_n$. a) Show that $S_n$ is $\Gamma(n, 1/\lambda)$. Each $X_i$ is $\Gamma(1, 1/\lambda)$ by the ...
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1answer
105 views

Can a two-sample t-test be used with data that doesn't follow a normal distribution?

One of the assumptions for t-tests is that the data must follow a normal distribution. However, due to the Central Limit Theorem (and this thread): "if the sample is large enough you can use t-test ...
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2answers
170 views

How useful is the CLT in applications?

My lecturer just covered the Lindberg-Levy central limit theorem and the multivariate version, the Lindberg-fuller CLT. I understood the basic concept and I can derive it, etc. But it would help my ...
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1answer
68 views

What does it mean to scale random variables?

We just started learning asymptotic theory, and to prove the lindberg-levy central limit theorem, weak law of large numbers etc, we 'scale and standardize' the RVs so it ends up having a standard ...
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24 views

Probability of an SRS having a mean between two values

Can anybody help me out with the following question? The distribution of the age at which all married males got married is right skewed with mean = 22.9 years and  standard deviation = 1.5 years. ...
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1answer
66 views

When Monte Carlo simulation can't be used to simulate a statistical system?

My question is simple. Which are the general conditions for which a Monte Carlo simulation can be used to represent a statistical system? Or conversely, which are the statistical system that cannot be ...
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1answer
71 views

Law of Large numbers and central limit theorem

I have come across a bit of a dilemma in my exam revision, this is not a topic I am particularly strong with so help is most appreciated: Assume that $X_1,X_2,...$ is an i.i.d sequence, where ...
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1answer
88 views

Does the central limit theorem imply the law of large numbers

Assuming that the distribution has finite variance (a condition not required for the LLN), then doesn't the LLN follow from the CLT?
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1answer
65 views

What characteristics of the distribution of a test statistic can be inferred using a bootstrap?

Setup: Let $p_n(x) = \mathbb{P}(S_n \leq x)$ and let $p_n^*(x) = \mathbb{P}^*(S_n^* - S_n \leq x)$, where $S_n$ is some zero-mean test statistic that can validly be bootstrapped, eg a sample mean, and ...
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196 views

When combining p values, why not just averaging?

I recently learned about Fisher's method to combine p-values. This is based on the fact that p-value under the null follows a uniform distribution, and that $$-2\sum_{i=1}^n{\log X_i} \sim \chi^2(2n), ...
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42 views

Why does the accuracy of the central limit theorem for a mean depends on the skewness of the random variables being summed?

The accuracy of the central limit theorem for a mean depends on the skewness of the random variables being summed. Could someone explain that to me? Or could someone propose a book to read? ...
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62 views

Central Limit Theorem - interval estimation

I'm rolling a regular dodecahedron (12-side die) 1200 times. I need to find an interval, in which the total count of prime-number results will lie with the probability of 95%. I have to use the ...
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48 views

CLT in a Monte Carlo simulation, small sample

A CLT says that asymptotically the sampling distribution of the sampling mean converges to the Normal. I would like to run a Monte Carlo simulation using information on one of the model's variables ...
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101 views

Distribution of a sample from a normal population

If I were to collect ONE sample of size 20 from a normal population, how justified would I be in claiming that the sample is normally distributed? I'm getting a little confused since by the Central ...
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1answer
72 views

Rate of convergence for SLLN

I am interested in writing a non-asymptotic rate of convergence for SLLN as a function of number of samples. From the literature I've read so far, CLT provides an asymptotic convergence rate of ...
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1answer
59 views

Is there a statement for the median like the central limit theorem for the mean? [duplicate]

According to the Central Limit Theorem (CLT) in statistics, the distribution of the average of randomly sampled n observations tends to follow normal distribution as the sampling size n becomes ...
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2answers
120 views

Taking the log of variables

Just before I start the question I would like you all to know that I have checked the other threads on taking the log of variables but I still think I have a question that hasn't been touched on yet. ...
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1answer
26 views

Can the selling price be treated as a random variable?

I have some data from my company that I have been looking at. I have been comparing deals that we have "lost" to deals that we have "won". I have been comparing the product averages and standard ...
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1answer
91 views

Convergence in distribution

For a statistic $T_n = \frac{1}{n} \sum_{i=1}^nY_i - \frac{1}{a}$. Prove directly (without CLT) that scaled and appropriately shifted version of $T_n$ converges in distribution to $N(0,1)$. [EDIT] ...
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45 views

Does the sum of several distributions become more central or approximated to Normal

As the classic CLT states Xs follow the same distribution, then the sum of Xs approximate to Normal distribution. But what about several Xs follows different distributions (maybe the same class but ...
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179 views

Cauchy Distribution and Central Limit Theorem

In order for the CLT to hold we need the distribution we wish to approximate to have mean $\mu$ and finite variance $\sigma^2$. Would it be true to say that for the case of the Cauchy distribution, ...
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137 views

Asymptotic normal distribution via the central limit theorem

I have a sample $n = 100$ with two "successes" (Two kids having a disease among 100). So we obviously have a binomial distribution. First I had to compute the maximum likelihood (ML) estimator ...
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97 views

Explaining why process obeys Central Limit Theorem

I'm trying to explain why some complex process obeys Central Limit Theorem. The process is a chip compiler that runs complex place & route algorithms. The input is an integer seed. It initializes ...
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35 views

CLT for sum of products of Gaussian variables

I have two multivariate Gaussians $\mathbf{x} \sim \mathcal{N}(\mu_x, \Sigma_x^2)$ and $\mathbf{y} \sim \mathcal{N}(\mu_y, \Sigma_y^2)$. I assume that the covariances are diagonal. I am interested in ...
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42 views

Application of central limit theorem

I'm wondering if I could use an CLT theorem to show that $S_n = \sum_{i=1}^n a_i X_i,$ converges to a normal distribution for large $n$, where $a_i$ are real constants and $X_i$ are I.I.D. poisson ...
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41 views

Stratified Sampling and the Central Limit Theorem

What can be said about the convergence rate of stratified sample means to a normal distribution, given different allocation schemes? Obviously, under very poor allocation, this convergence can fail ...
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60 views

Is there anything similar to Central Limit Theorem for Variance for Hypothesis Testing?

Central Limit Theorem says, "given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and ...
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45 views

Why can't I use the variance of the sample average in the Central Limit Theorem for the weak-stationary process?

Under mild conditions $\dfrac{\bar{X}-\mu}{\sqrt{\sigma^2/n}}$ approaches the standard normal (where $\sigma^2$ is the process variance, not the marginal variance $\sigma^2_x$). Why is the ...
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67 views

The Central Limit Theorem in Quantile Estimation

I don't have a very strong background in statistics so I have a few conceptual questions and there is a strong possibility I'm missing something obvious. Suppose I'm interested in estimating the 99th ...
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2answers
91 views

If I reject the null on a two sample t-test, can I conclude which direction the difference goes?

Since it's a two-sided test, it seems plausible that I can only reject that the difference is 0. However, if the entire interval is negative or positive can I say that one is larger or smaller than ...
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532 views

Example of distribution where large sample size is necessary for central limit theorem

Some books state a sample size of size 30 or higher is necessary for the central limit theorem to give a good approximation for $\bar{X}$. I know this isn't enough for all distributions. I wish ...
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52 views

6-sided die roll average value [duplicate]

A die is rolled 100 times. Find the probability that the mean value would be between 3 and 3.5. I've found the expected value which is 3.5 without idea what to do next.
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41 views

Does the central limit theorem hold for the prediction error over different samples?

Given an (infinite) data population from which you repeatedly draw samples of a fixed size. On each sample you learn a classifier which you then evaluate by computing the prediction error on a large ...
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54 views

Deriving a “confidence relationship”?

I understand the basics of confidence intervals, the central limit theorem, etc, to be able to know things like given N samples of random variable, we're 68/95/99.7 percent sure the variable is within ...
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121 views

Confidence intervals based on the CLT: ever useful?

Suppose, for concreteness, that I am trying to estimate the mean of a population using a random sample of size $N$. Many elementary books discuss forming a confidence interval for the population mean ...
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143 views

Use the Central Limit Theorem to calculate the approx probabilities of Gamma RVs?

If $X_i, i=1,...,$ are independent, identically distributed $\operatorname{N}(0,1)$ random variables, and $Y_i = X_i^2$ are independent $\operatorname{Gamma}(\frac{1}{2},\frac{1}{2})$ RVs, use the ...
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357 views

What's the distribution of $\bar{X}^{-1}$?

What's the distribution of $\bar{X}^{-1}$ with X being a continuous iid random variable that is uniformly distributed? Can I use the CLT here?