"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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How is Central Limit Theorem affected when hypothesis space is restricted?

Some people told me that Central Limit Theorem and other convergence theorems break when we restrict space of distributions to those that are effectively computable. How true is this statement, and ...
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47 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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20 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
37 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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162 views

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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53 views

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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45 views

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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16 views

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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67 views

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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74 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
103 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
121 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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53 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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29 views

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
511 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 ...
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1answer
37 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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26 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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4answers
68 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
121 views

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
37 views

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 ...
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1answer
85 views

Sample size and Central Limit theorem

I'm working on an example in introductory statistics and I'm not sure if my answer is correct. I think 2,3,5 is correct and not sure about 1,4. Am I correct? It's about choosing correct statements ...
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1answer
97 views

Central limit theorem question

I am thinking about the rate of convergence in central limit theorem (CLT) for different distributions. Let's assume we have a set of i.i.d random variables, $X_1,X_2,\ldots$ which follow an unknown ...
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3answers
69 views

Central limit theorem for sum of differences

I have a quick question. If we have a sequence of i.i.d random variables, $X_1,X_2,X_3,\ldots,X_n$. Then, we make sequence below: $$ S=X_1-X_2+X_3-X_4+\ldots+X_n $$ Does central limit theorem work in ...
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Number of trials nesessary to determine probability with given credibility

I have the following problem, "A web page contained a link with CTR=1.5%. The link has been modified: now it appears in 4% of all page shows. What number of impressions of the page do we need to ...
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94 views

Question about normality assumption of t-test

For t-tests, according to most texts there's an assumption that the population data is normally distributed. I don't see why that is. Doesn't a t-test only require that the sampling distribution of ...
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Why is normality assumption so important (even for large N) for Chi-square test for the Variance

In my textbook i found the note that for a Chi-square test ($\chi = \frac{(n-1) \cdot s'^2}{\sigma^2}$) the assumption of a normal distribution in the population is very important - and much more ...
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99 views

Central limit theorem: applicability for assumptions of different tests

Since many statistical procedures (e.g. t-test, ANOVA, Pearson’s r (for efficient estimates)) require the normal distribution of the tested variables ('normality-assumption') one may ask if (at least) ...
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46 views

Error propagation - nonnormal (again)

I have a dataset of ~2000 points. Each of those points has a standard error value associated with it, and it is assumed that the data points and errors are uncorrelated. Both the dataset and the ...
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1answer
57 views

Generalized linear models and central limit theorem

If a comparison of treatment means can be made with ANOVA or GLM because it is assumed errors are normally distributed as suggested by the central limit theorem, why would it be necessary to implement ...
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56 views

Central-limit theorem via sample size or sampling magnitude?

I have a small application of the central limit theorem: "Take a sample of 40 exponentially distributed random numbers and calculate their mean. Repeat 1000 times, record each measured mean and ...
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22 views

T-test for a small sample with unknown distribution [duplicate]

Consider a simple hypothesis test concerning the mean of a single sample. If the sample is normally distributed and the variance is known, the exact distribution of the sample mean is known ...
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1answer
173 views

Convergence from Gamma to Normal Distribution

I came across this problem: Problem If I have $X_1, X_2, ..., X_n$ $n$ iid random variables which pdf is $$ f_X(x) = \begin{cases} \dfrac{x^{\mu-1} e^{-x}}{\Gamma{(\mu)}} &0<x<\infty, ...
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Definition of random walk as a summation of independent random processes

I have a complete beginner question on random walk. As per this paper ...
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1answer
54 views

Asymptotic distribution for moments of gaussian distribution

Is there a way to find the asymptotic distribution for the moments of Gaussian distribution? More specifically, say you have $X_1, ..., X_n \sim N(\mu, \sigma^2)$. For a moment $m_{n, k} ...
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62 views

Non parametric test after multiple imputation

In advance, I'm not a statistician and I'm a SPSS user (although I also use STATA fairly well and R rarely). I'm having some trouble in making some nonparametric comparisons after Multiple ...
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1answer
72 views

Convergence of sequence of random variables

$X$ is the number of heads we got after tossing $n$ fair coins. My question is: $P(X< \frac{n}{2}+\sqrt{n})$ if $n \to \infty$? I tried to apply CLT like this: ...
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central limit theorem applications in risk management

Is there an example of the central limit theorem being used in financial markets, preferably in the field of risk management? I am not able to find applications to financial markets for this ...
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923 views

Reasons for data to be normally distributed

What are some theorems which might explain (i.e., generatively) why real-world data might be expected to be normally distributed? There are two that I know of: The Central Limit Theorem (of ...
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11 views

Reasons for data to be normally distributed [duplicate]

What are some theorems which might explain (i.e. generatively) why real-world data might be expected to be normally distributed? There are two that I know of 1) The Central Limit Theorem (of ...
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211 views

What is the limiting distribution of the sample mean?

My question is relatively simple: what is the limiting distribution of the sample mean? But there are some technicalities I want to discuss. context: I was asked this problem in an exam, and I feel ...
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1answer
96 views

Do these random variables satisfy Lindeberg's condition?

I have the followig sequences: $Pr(X_n=n)=Pr(X_n=-n)=0.5$ $Pr(X_n=2^{n/2})=Pr(X_n=-2^{n/2})=0.5$ I have to show whether they satisfy Lindeberg's condition or not, but this condition is a bit ...
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Upper bound for sampling fraction for CLT to hold

What is the upper bound for the sampling fraction for the Central limit theorem to hold when sampling without replacement? Context The context for my question is that it is regularly argued (see e.g. ...
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4answers
90 views

Demonstration of central limit theorem

I teach basic (very) statistics to prisoners in a medium/high security prison and would like to demonstrate the Central Limit Theorem. The classroom has no resources beyond a white board. I can only ...
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2answers
106 views

Does the central limit theorem apply to these probability density functions?

Let's say you have n uniform random variables from 0 to 1. The distribution of the average of these variables approaches normal with increasing n according to the central limit theorem. What if ...
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99 views

What is the multivariate analog of the median?

There is a univariate mean: sum the points and divide by the count. There is a multivariate mean analog - the centroid, a point in a multidimensional space. (1). For the median one sorts the list ...
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1answer
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mean of population with repeated samples of varying size - how to apply CLT?

I have a population with mean $\mu$ and variance $\sigma^2$. I draw a sample with $n_i$ number of i.i.d random observations from population. I compute the mean for this sample as $mean_i$. I then ...
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43 views

Expected value non-independent random variables

Let $X$ be a set of costumers, {$x_1, ..., x_N$}, each $x_i \in X$ have a discount $p_i$ in the interval $[0,1]$, it means if $p_i$ is 0.3, $x_i$ will pay only 0.3 of the entire value. I want to know ...
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50 views

Variance in central limit theorem

Why is it that $\sqrt{n}(X_{n}-\mu)$ converges in distribution to $N(0,\sigma^{2})$ but $\sqrt{n}(X_{n}-\mu)/\sigma$ converges in distribution to $N(0,1)$?
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Expectation of ratio of functions of Bernoullis: a concentration question

Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$, and let ...
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59 views

A simple question on CLT in possible connection with Berry-Esseen thm

I am curious about the contents while I read a note on machine learning. It could be obvious. So, please let me know if I am missing some fundamental things. $X_1,X_2,...,X_n$ are from an i.i.d. ...