"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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37 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} ...
0
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0answers
8 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 ...
2
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1answer
32 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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0answers
16 views

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 ...
14
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4answers
571 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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0answers
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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0answers
42 views

Central/limit theorem problem

Lets have a sum of 500 i.i.d. random variables. Each assumes values 0 with probability 0.3, 1 with probability 0.2 and 2 with probability 0.5. Find the smallest $m$ such that probability that the sum ...
3
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0answers
35 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 ...
4
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1answer
58 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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0answers
16 views

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. ...
5
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4answers
73 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 ...
4
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2answers
84 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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52 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 ...
2
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1answer
47 views

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 ...
1
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0answers
33 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 ...
1
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1answer
41 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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0answers
15 views

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 ...
0
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1answer
35 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. ...
1
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2answers
267 views

Normal approximation for large data set?

I have a dataset that is highly skewed. See image below: When I transform the data I get the following histogram that makes it look normal: This data however is not normal. I get a p-value of ...
3
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0answers
54 views

How to find an appropriate sample size given information about the CV?

You are planning to collect a (simple random) sample to estimate the mean of a non-negative random variable. It is known that the population coefficient of variation (CV) is 1.2. Use the central ...
3
votes
1answer
167 views

Central Limit Theorem, what does it really say? [duplicate]

I know that the Central Limit Theorem states that if a random sample of size $n$ is drawn from iid random variables $X_1, \ldots, X_n$, then the variable $$ Z = \frac{\hat x - E(X) }{{\rm sd}(\hat ...
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0answers
41 views

ANOVA's and reaction time data

I've been reading a lot on the effect of perceptual load on selective attention but the analysis confuses me somewhat. Some of these studies flag 20-40ms differences between conditions as significant ...
4
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2answers
60 views

Sampling distribution of the mean of population that assume values only between 0 and 1

I'm trying to use central limit theorem to compute the mean of the sample, however the population where I'm sampling has value only between 0 and 1, can I use mean of the sample as mean of the ...
0
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0answers
10 views

Normal distribution approximation using central limit theorem [duplicate]

I have a variable with 150 observations and it is non normal. Wouldn't CLT be useful in this context. What does large n really mean in CLT?
3
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1answer
348 views

Confidence interval for the standard deviation on a bimodal distribution

I have a bimodal distribution, and I wish to estimate the mean and standard deviation of the population (well, these could be 2 sub-populations according to the shape). With the mean I have no ...
2
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0answers
56 views

Standard error on median for exponential distribution

I am trying to find the standard error on the median, $\sigma_\tilde{x}$, for a sample, $X_i$, of a population whose pdf could be modelled as $\lambda e^{-\lambda x} $ if normalized. To make sure we ...
5
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1answer
49 views

Reference request: local limit theorem for log-concave densities

The following is easy to prove and can't possibly be new. But I can't find it printed anywhere despite some effort. Can anyone tell me where it is published? Let $X_1,X_2,\ldots$ be a sequence of ...
6
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6answers
440 views

Dealing with non-normal distribution in “big” datasets, when do we throw out the CLT?

Apologies from the go as this question comes from an absolute newbie and will definitely not satisfy a lot of the detail required. Hence, your guidance in providing you the right information to allow ...
6
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1answer
178 views

Asymptotic distribution of sample variance of non-normal sample

This is a more general treatment of the issue posed by this question. After deriving the asymptotic distribution of the sample variance, we can apply the Delta method to arrive at the corresponding ...
3
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1answer
207 views

Question about standard deviation and central limit theorem

I have a quick question about the central limit theorem. Lets say I measure some value that comes from an arbitrary distribution N times and I repeat this M times. I understand that if I calculcate ...
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0answers
45 views

Probability that a sample mean is between two values using Central Limit Theorem

Mean is $2.707$, standard deviation is $.049$, sample of $35$ is drawn from the population. What is the probability that the mean price for the sample was between $2.683$ and $2.716$? It was suggested ...
0
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1answer
66 views

Quick Question - Approximate distribution for sample mean?

I am having issues answering part two. I think it is about CLT, correct me if I'm wrong. But how do you compute the distribution details from this ? Please help and thank you for all your ...
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1answer
27 views

Use of standard error of the mean when the components are not identically-distributed

Let {$X_1$, ..., $X_n$} be a random sample of size n in which the different elements are measurements drawn from $m$ different populations with different distributions. From the Central Limit Theorem, ...
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2answers
74 views

How do I test that a sequence of data satisfies the Central limit theorem?

I have some data stored as a list from some computations. I want to calculate the "distance" the sequence is away from satisfying the central limit theorem. I would like to know, what should I use as ...
0
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0answers
89 views

Error for combining multiple binomial distributions

This problem is somewhat involved and I have a partial solution so bear with me. I will illustrate the problem with an example. Lets say we have two processes and we want to know which has a higher ...
7
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1answer
134 views

Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?

Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a ...
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2answers
80 views

Self-study question on CLT

Good evening everyone, I am currently doing a self-study on CLT and was working on an exercise which asks if the following statement is true or false CLT guarantees that the population mean is ...
2
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2answers
89 views

Convergence in distribution results not deriving from central limit theorem?

In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem. Out ...
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2answers
73 views

Normal Distribution CLT Question

I am working on a self-study question where A study indicates that the typical American woman spends USD 340 per year for personal care products. The distribution of the amount follows a ...
0
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1answer
49 views

Distribution of a Mean and Variance

Say we have observations $x_1 \dots x_n$ and we have some sort of Bayesian framework where we would like to estimate a distribution for the mean $\mu$ of our observations and the variance $\sigma^2$ ...
8
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3answers
184 views

If $Z_i =\min \{k_i, X_i\}$, $X_i \sim U[a_i, b_i]$, what is the distribution of $\sum_iZ_i$?

Assume the following set up: Let $Z_i = \min\{k_i, X_i\}, i=1,...,n$. Also $X_i \sim U[a_i, b_i], \; a_i, b_i >0$. Moreover $k_i = ca_i + (1-c)b_i,\;\; 0<c<1$ i.e. $k_i$ is a convex ...
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1answer
58 views

Theoretical Justification for Cross Validation

I get it, cross validation works. I'm wondering if there is an literature out there giving any theoretical justification for cross validation. My thought is that there should be, at least, something ...
5
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1answer
146 views

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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1answer
88 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
50 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
79 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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204 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 ...
6
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1answer
131 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 ...
2
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1answer
108 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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1answer
107 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 ...