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Questions tagged [central-limit-theorem]

"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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What will be the confidence interval(for below question) for 99 percent confidence level [68 95 99 rule] [on hold]

Is the below solution correct....because the probability of obtaining a sample which would have the mean height of 70 will fall under the intervals calculated using the mean+/- 3(SD) [99% Confidence ...
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Central limit theorem for resampling

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic? As a simple example, suppose there are $n$ i....
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Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
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central limit theorem: do we care about standard deviation within one sample of size n? [duplicate]

I'm learning applications on Central Limit Theorem and got really confused with a few points. Think of an example of applying Central Limit Theorem: We have a whole population of 10 billion items ...
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Do we use the SD of whole population or SD of just one sample to calculate SE of samples means in central limit theorem?

I'm learning applications on Central Limit Theorem and got really confused with a few points. According to this tutorial, the procedure to apply CLT usually goes like this: So if ...
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Self-study: Which option is NOT a prediction of the central limit theorem?

"All of the following are predictions of the Central Limit Theorem except: 1) The sample mean distribution will be approximately normally distributed if the sample size is large 2) The mean of the ...
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Understanding regression to mean

As far as I can understand regression to mean is that if I make a measurement, say the mean of the test scores of some students, when the measurement is repeated with the same student with same ...
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How to find the “variance” when using “central limit theorem” on a Poisson distribution? [closed]

Assume we have N number of inventors in a company. Inventor i expects to invent X_i number of inventions per year. How many inventions each of them invent per year has a "Poisson distribution" where ...
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Asymptotic distribution of median estimator when density doesn't exist

We know that when density(say $f$) exists at the median(say $\theta$) then the median estimator(say $\hat{\theta_n}$) has the following property $$ \sqrt n(\hat{\theta_n}-\theta) \to^d N(0,1/\{4f(\...
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The IQs from 181 boys aged between 6-7 years old were measured. Calculate its mean's confidence interval for $\alpha = 5\%$

The IQ from 181 boys aged between 6-7 years old were measured. The mean IQ is 108.08, and the standard deviation is 14.38. (a) Determine the confidence interval with confidence coefficient $95\%$ ...
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Proof of the multivariate Central Limit Theorem

Casella and Lerner's "Theory of Point Estimation" (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given.  What would be its derivation?
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The limit distribution of Wilcoxon signed rank statistic?

An alternative representation of the Wilcoxon signed rank statistic $V$ is $V=\sum_{i\le j}\mathbb{I}_{\{X_i+X_j>0\}}=\sum_i\mathbb{I}_{\{X_i>0\}}+\sum_{i<j}\mathbb{I}_{\{X_i+X_j>0\}}$ ...
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Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem?

The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
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Central Limit Theorem and hypergeometric distribution

We have a game V, which is about pulling 3 cards repeatedly from a deck of 52 cards. For each card with a picture (12 of 52 cards are with a picture) then we win 100 USD of money for each card. We ...
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How to derive the asymptotic distribution of t-statistic?

Let ${X_n}$ be an IID sample such that ${X_i} \sim N(\mu,\sigma^2)$. When both $\mu$ and $\sigma$ are unknown, we construct $t(\hat{\mu},s)=\dfrac{\sqrt{n}(\hat{\mu}-\mu)}{s}$, where $s$ is the sample ...
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Confused about Cramer-Rao lower bound and CLT

Learning statistics for application in the physical sciences. I am confused about the cramer-rao (CR) bound vs central limit theorem for estimating the variance of the sample mean. I thought that once ...
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T-Test Raw Data vs Sampling Data with CLT

I have some data that I want to run a T-Test on and get a p-value of 0.05. See below: I then using the CLT take 30 samples from each sample and take the mean of each sample. I do this 100 times to ...
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Is it the case/is there a proof that the convergence in distribution for the CLT is monotonic?

So for instance, if I compare $\bar{x}_n$ and the comparable normal distribution, and $\bar{x}_m$, $m > n$, and the comparable normal distribution, would I expect the difference in former (e.g. the ...
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What process could lead to a sampling distribution of the mean that is approximately skew normal

I have a data set representing guest ratings of various hotels. Unfortunately, I don't have access to the ratings given by individual guests, only to the mean guest rating for each particular hotel. ...
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Why does Central Limit Theorem break down in my simulation?

Let say I have following numbers: 4,3,5,6,5,3,4,2,5,4,3,6,5 I sample some of them, say, 5 of them, and calculate the sum of 5 samples. Then I repeat that over ...
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Asymptotic Expectation of Ratio of Sample Averages

I have two random variables: $X$ and $Y$. I know that: \begin{equation} E[X]=E[Y]=\mu>0 \end{equation} I know that variance of both can be bounded: \begin{equation} \operatorname{Var}[X]<k, \...
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Terminology: “Central Limit Theorem” for Delta Method

This is a question about when is it appropriate to call an asymptotic normality statement, the "Central Limit Theorem" (CLT). More specifically, suppose I have $X_1, X_2, \dots X_n$ iid from a ...
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Central limit theorem doubt [duplicate]

Consider a random variable $X$ that takes only positive values, (in my case the r.v. $X=Y^2$ where $Y$ is a random variable itself). We know that from the CLT we have that $\sqrt{n}(\bar{x}-\mu)\...
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proper sample size for the central limit theorem to hold

I have tested the central limit theorem with 1000 samples and a sample size of 4. The resulting distribution was nowhere near normal, and when I used n = 30, it did start looking normal. Is there a ...
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CLT for uniform distribution

I don't understand how the CLT can hold for a uniform distribution. Say I have U[0;1], then whatever value I will be able to sample from the population will always be 1. Therefore, every sample mean I ...
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Convergence of poisson distribution

Let $X\sim \operatorname{Pois}(\lambda)$ and $x_1,\ldots,x_n$ observations following this distribution. I want to derive the analytical solution of the following series: $$\ell(\lambda):=\lim_{x\...
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1answer
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Using CLT for hypothesis testing

I have two not normal distributions (~1k samples in each), looks like exponential: So, I need to check its means, that's why I have following questions: The easiest way to do it - is to use Mann-...
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Why does increasing the sample size of coin flips not improve the normal curve approximation?

I'm reading the Statistics (Freeman, Pisani, Purves) book and I'm trying to reproduce an example where a coin is tossed say 50 times, the number of heads counted and this is repeated say 1,000 times. ...
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Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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Distribution for average of multiple binomial proportions

Assume we have a population $N$ and a proportion $p$ of that population with a characteristic of interest. Both $N$ and $p$ are unknown. Furthermore, assume that we have $k$ random samples $(n_i, x_i)$...
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How to calculate the mean of a sub sample, given the mean of the super sample and the standard deviation of the population?

I need to run a simulation of cash flows for a project. We are selling a service. The service comes with a range of options. Depending on the specific options chosen, the cost of the service can ...
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Central limit theorem (CLT) writing

Is there a reason why we are used to write the CLT as $\sqrt{n}(\overline{X}_n-\mu)\stackrel{d}{\rightarrow}N(0,\sigma^2)$ and not as $\overline{X}_n\stackrel{d}{\rightarrow}N(\mu, \frac{\sigma^2}{n})$...
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Parametric Bootstrap Central limit theorem non i i d

I am having paired data with missing values in a single arm. I am willing to use parametric bootstrap with specific quadratic tests to test the hypothesis of equality of means. My model is as follows:...
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Is the distribution of the logarithm of the mean of Bernoulli random variables ($\log \overline X$) still asymptotically normal?

Let $\overline X$ be the mean of a Bernoulli random variable (r.v.) $$\overline X = \frac{1}{n}\sum_{i=1}^{n} X_i$$ where $X_i \in \{0, 1\}$. So based on Central Limit Theoreom, $$\overline X \sim \...
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What is the relationship of skew and sample size on the sampling distribution?

I'm interested in the relationship between a distribution's skew and the sample size needed for the sampling distribution to be approximately normal. Let's assume the distribution is unimodal. In ...
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Decreasing profit margin in relation to cost price

Is there a formula that can adjust the profit margin by decreasing it the larger is the cost price amount. For example: If I set a profit of 50%. I will resell an item that cost 50 $ for 75. While ...
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Sufficient Conditions for the Central Limit Theorem

My understanding is that the central limit theorem applies as long as the variance of the random variable is less than infinity. Is this equivalent to saying that all moments are finite? If not, what ...
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Stationary processes that do not satisfy Gordin's central limit theorem

We are doing an assignment for our Advanced Econometrics course for which we are trying to illustrate Gordin's Central Limit Theorem by simulation. We used an AR(1) process to show that if the ...
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1answer
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jackknife estimator with central limit theorem

Let $\hat{\theta}_n$ be an estimator of the parameter $\theta$ from the sample $\Omega_n$ of $n$ observations, satisfying that $\sqrt{n} (\hat{\theta}_n-\theta) \overset{d}{\longrightarrow} \mathcal{N}...
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CLT and convergence of Variance

I am looking at a problem where the sum of the individual $X_i$ is $S_n=X_1+\dotsm+X_n$. The probability is given as, $P(X_i=i)=P(X_i=-i)=\frac{i^{-\alpha}}{4}$ and $P(X_i=0)=1-\frac{i^{-\alpha}}{2}$. ...
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Confidence interval for $\sigma^2$

I started with any distribution and underwent the CLT on $\sqrt{n}(\widehat{\sigma}^2 - \sigma^2)$ where $$ \widehat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \mu)^2 $$ is a sample mean of $\sigma^2$...
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Question about CLT proof

I'm working through the CLT proof on Wikipedia trying to get a better intuition, and it made me wonder what an individual distribution looks like after dropping the o(t^2/n) terms from the Taylor ...
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Can the Berry-Esseen theorem tell us whether acceptable inference may be achieved by parametric tests?

I refer in particular to such choices: 1) t-test (or its generalizations: ANOVA or Hotelling's $t^2$) vs its non-parametric alternatives (e.g., U Mann-Whitney test and its generalizations); 2) Pearson'...
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CLT approximation - how large should sample be so probability is equal to 0.95? [duplicate]

We have a measurement which has mean $\mu$ and variance $\sigma^2$ = 25. Let $\bar{X}$ be average of $\textit{n}$ such independent measurements. How large should $\textit{n}$ be in so that $P(|\bar{...
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Do all sample statistics behave normally?

A version of the central limit theorem tells us that the sample means will be distributed roughly like a normal distribution around the population mean. Are there cases of a sample statistics that ...
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If I roll a die 25 times what is the probability that the mean outcome is greater than 4?

Consider a problem: You take a fair die to a party and announce that you will roll it 25 times. You will record each outcome and at the end average the 25 outcomes together to get their ...
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1answer
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T-test or Mann-Whitney U test [duplicate]

Given the central limit theorem, can you always use a t-test to test a difference between two groups even if the data are not normally distributed but the sample size is large enough? Stated in ...
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normalization coefficient in the central limit theorem [duplicate]

why do we use normalization coefficient in the central limit theorem? For CLT we use $\sqrt{n}$ as the normalization factor, but why do we need it?
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Trouble relating the Central Limit Theorem to confidence intervals

I'm having trouble understanding how the Central Limit Theorem (CLT) implies that we can create confidence intervals as we do. For example, Slide 5 from these lecture notes essentially lays out the ...