"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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Why is this definition of the Central Limit Theorem not incorrect?

I found the following definition of the Central Limit Theorem from book Probability and Statistics by Degroot (also from Wikipedia). It simply states the CLT as $$ \lim_{n\to\infty} ...
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Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
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254 views

Central limit theorem and the Pareto distribution

Can somebody please provide a simple (lay person) explanation of the relationship between Pareto distributions and the Central Limit Theorem (e.g. does it apply? Why/ why not?)? I am trying to ...
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41 views

Central Limit Theorem for Wilcoxon signed rank tests

Let $X_i, i=1,2,...,n$ be a set of iid observations, assumed symmetric about $\mu$. Let $R_i$ be the rank of the absolute deviations from some $\mu_0$, i.e. $R_i=\text{rank}(|X_i-\mu_0|)$. Let ...
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35 views

Relationship between CLT for estimator of sample and number of combinations for subsample?

What is the relationship between the Central Limit Theorem as applied to the expected value of an estimator of a parent sample and the number of possible combinations of a subsample used to calculate ...
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145 views

Two Sample chi squared test

This question is from Van der Vaart's book Asymptotic Statistics, pg. 253. #3: Suppose that $\mathbf{X}_m$ and $\mathbf{Y}_n$ are independent multinomial vectors with parameters $(m,a_1,\ldots,a_k)$ ...
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18 views

A conditional normal random variable conditioned on a normal distributed random variable

I am trying to solve the following problem. Given $S \sim N(\mu_S,\sigma_S^2)$ and $U| S=s \sim N(s p,sp(1-p))$, what is the distribution of $U$? By the central limit theorem (CLT), the conditional ...
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Exercise on Chebyshev inequality compared to the Central Limit Theorem

Problem Take this (easy) problem as an example: An astronomer is interested in measuring the distance, in light-years, from his observatory to a distant star. Although the astronomer has a measuring ...
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Is finite and strictly positive variances all that is needed for the CLT Lindeberg condition?

I cannot understand what I am doing wrong here, so please somebody, point it out from me. The issue? I keep finding that the Lindeberg sufficient condition for the Central Limit Theorem for ...
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38 views

Is there a central limit theorem for i.n.i.d. variables when normalised by inconsistent variance estimate?

I am wondering whether there exists a central limit theorem for the following situation. Consider the sum of normally distributed variables $\epsilon_i$ with unequal variances according to ...
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117 views

Topologies for which the ensemble of probability distributions is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess. ...
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1answer
140 views

Convergence to normal distribution

Let $X_{1},X_{2},...$ be independent random variables such that $X_{k}$ is Po(k)-distributed for k=1,2... Show that: $$Z_{n}=\frac{1}{n}\sum_{k=1}^{n}\left(X_{k}-\frac{n^{2}}{2}\right)$$ ...
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29 views

distribution of $T^2$ statistic without Gaussian assumption

I would like to know if the Gaussian assumption is needed on $x_i$ in deriving the asymptotic distribution of the $T^2$ statistic. Here is the presentation sequence I got from the Wiki page on ...
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Confidence intervals and Central Limit Theorem with only one sample

I know that to erect confidence intervals, standard errors must be calculated, a process which in turn makes use of the CLT (but I am not clear how). I also understand that, very generally, the CLT ...
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10 views

Central Limit Theorem- observations vs samples [duplicate]

I'm confused with the concept of central limit theorem and I need some clarification about it, hope you guys can help! Now suppose there is a population in form of numbered balls contained in a box. ...
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28 views

Difference between asymptotic normalities of OLS and MLE

Let's see the comparison below Asymptotic normality is given by CLT for both cases. In MLE case, a variance of $\hat{\theta}$ is in distribution as $\frac{1}{I(\theta)}$, but in OLS case ...
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17 views

Fisher information of multiple samples and the standard error

I'm curious on whether the two following results are related (or indeed just the same result). First result The Fisher information is some measure of how well one can estimate the value of a ...
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1answer
86 views

Central Limit Theorem for R coding assignment

I had a homework assignment asking us to code some basic things in R. Here are the tasks: Generate a Gamma(1,2) population of size N=5000 and show its density plot and descriptive statistics. Create ...
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63 views

Sum of random variables without central limit theorem

I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution ...
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Finding n using central limit theorem

Problem 3: Central Limit Consider tossing a fair coin $100n$ times, where n is geometrically distributed with mean 6. Due to a counting error, it is reported that the fraction of heads is in ...
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1answer
43 views

Do we draw samples with or without replacement when we state the central limit theorem

In central limit theorem (in its most basic flavor), , we say that we draw a large number of samples from an unknown distribution and each time we calculate the mean. Then, for a large sample size, ...
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Clarification on central limit theorem

I found this definition of central limit theorem in the book: Intro to probability and statistics using R: I thought the sample mean $\bar X$ itself followed the normal distribution and not the ...
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1answer
32 views

Can I use sub-sample means to verify that the sample mean is approximately normal?

I have a large (N=10^6) Monte Carlo ensemble of runs of my engineering code. I want to compute a confidence interval for the means of two output variables, and falsify the hypothesis that they have ...
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Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
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Required number of random numbers for using Central Limit Theorem

I wanted to know how many i.i.d random variables have to be summed in order to be able to use Central Limit Theorem. I know it varies depending on the distribution, but does there exist any number ...
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150 views

The role of variance in Central Limit Theorem

I've read somewhere that the reason we square the differences instead of taking absolute values when calculating variance is that variance defined in the usual way, with squares in the nominator, ...
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42 views

Local Version of Bernstein Von-Mises Theorem?

The Bernstein-Von Mises theorem says that, under reasonable conditions, the posterior distribution $p(\theta | x_{1},\ldots,x_{n})$ converges weakly to the normal distribution after suitable ...
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Why is there a need for a 'sampling distribution' to find confidence intervals?

I understand the key principles behind confidence intervals, but there's something I want a bit of clarification on. Let's say I have a basket of apples that I picked at the orchard. The weight is ...
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51 views

Conjecturing asymptotic normality for a sum of dependent random variables

I am hunting for the asymptotic distribution of a scaled partial sum of pair-wise equi-correlated, identically distributed continuous random variables $$W =n^{-\delta}\sum_{i=1}^nY_i(n),\;\; ...
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296 views

Central Limit Theorem for Normal Distribution of Negative Binomial

The question is: Explain why the negative binomial distribution will be approximately normal if the parameter k is large enough. What are the parameters of this normal approximation? I have ...
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1answer
57 views

Central Limit Theorem Equation

I was wondering what the difference is between $Z_n=\frac{S_n-n\mu}{\sigma \sqrt{n}}=\frac{X_1+\ldots+X_n -n\mu}{\sigma \sqrt{n}}$ and $Z_n=\frac{X-n\mu}{\sigma/\sqrt{n}}$? What is the benefit of ...
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32 views

Multivariate central limit theorem with unknown variance

Let $X_1,\ldots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment. If $d=1$, $\bar{X}_n=\frac{1}{n}\sum_{i=1}^n X_i$ and $S_n=\frac{1}{n-1}\sum_{i=1}^n ...
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1answer
137 views

CLT simulation with R

I am trying to run a simple CLT simulation with R. I want to create a vector with 10000 dice rolls, then take 100 means of samples of "n" size to see how when "n" increases, the distribution starts ...
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Lindeberg CLT for exponential independent variables

Crossposted in math.stackexchange: CLT for independent, but non-identically distributed exponential variables This problem is self-study for my qualifying exam. Problem Suppose $(e_n)_{n\ge 1}$ are ...
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About the central limit theorem and statistical testing

Wikipedia states that In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent ...
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CLT and 2 variables

Okay so there are 2 variables $D_i$ and $V_i$. Now $D= D_1 + D_2 + ... + D_N$ and $V = V_1 +.. +V_N$ Now I know the relationship is such that $E[D_i - a*V_i] = 0$ and $Var[D_i - a*V_i] = E[D_i]$ ...
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Central limit theorem (CLT), average over 80 students but sum up over 52 weeks [duplicate]

Because we took the average over 80 students but sum up over 52 weeks, I feel that it’s correct only under certain assumptions but I can’t pinpoint them. Are they the following? The distribution in ...
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Does central limit theorem (CLT) hold for non-integer values of $n$? [closed]

Why I think it might be correct: Search “not an integer” in http://www.math.uah.edu/stat/sample/CLT.html and you would find 2 occurrences of the author stating that CLT is valid even if $n$ is not ...
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174 views

Approximating the distribution of a linear combination of beta-distributed independent random variables

This question is related with these other two questions in Cross Validated, which has been already answered: Approximate the distribution of the sum of ind. Beta r.v Central limit theorem when the ...
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1answer
196 views

Central Limit Theorem for Exponential Distribution

I've spent so long, and I think I'm missing something super obvious. The question: Let $X_1,\ldots,X_5$ be five independent variables from the exponential distribution with mean $2$. Write the pdf ...
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1answer
136 views

Central limit theorem when the mean is not constant

I've heard that there is a soft version or interpretation of the CLT that says —in summary— that it can be applied also to a sequence of independent real-valued random variables that share the same ...
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68 views

Why does the normality of the coefficient's probability distribution follow from the normality of the errors in OLS?

So suppose after a simpple OLS regression we want to know what the chance(P-value) is that the Beta coefficient is 0 . First we assume that many random processes caused the errors ($\epsilon\!_i$), ...
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bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
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3answers
540 views

Why are these file sizes not normally distributed?

I have saved 10,000 webcam images and tallied their lengths. The lighting conditions were constant throughout the recording time. The probability distribution is shown here, with my best efforts at ...
2
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1answer
63 views

Central limit theorem on a linear combination

I am looking for the name and the formulation of a CLT variant that states that a linear combination of random variables with the same mean and standard deviation will converge under a specific ...
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37 views

Total probability distribution of multiple random lotteries

My question: Imagine $d$ identical lotteries. Each individual lottery picks a cost $c_{i}$ between $0$ and $1$. Picking a costs occurs with probability distribution $f(c)$. The total cost of these ...
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90 views

Poor sample measurement and the Central Limit Theorem

I have a fairly basic question about the Central Limit Theorem. I understand it in principle, but I like to know specifically what happens when there is poor measurement on the samples. Say for ...
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1answer
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Why isn't the sample distribution of r normal?

The sample distribution of $r$ is positively or negatively skewed if $\rho \neq 0$. However, according to the central limit theorem - the more samples you take from a population, no matter what shape ...
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1answer
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Convergence of standardized means of a Bernoulli variable / CLT

The Question Consider a binary random variable X that satisfies: $Pr(X = 0) = \theta \ \ \ $ and $Pr(X = 1) = 1−\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size ...