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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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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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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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208 views

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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1answer
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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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1answer
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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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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 ...
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Question on central limit theorem

I have a question regarding central limit theorem! I understand that as the sample size increases and gets large enough, the sampling distribution of the sample mean can said to be approximated by a ...
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Slutsky's theorem applied to Bernoulli random variables

Suppose $X_n$ is a sequence of $n$ Bernoulli random variables with unknown $p$, and I try to get a confidence interval for $p$. Using central limit theorem, I've got \begin{align*} \frac{\bar X_n - p}...
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What formula for a Confidence Interval of the difference in proportions when sample sizes are small

Suppose that we are interested in comparing two approximately normal sampling distributions described by random variables $ \displaystyle \frac{Y_1}{n_1} = N(p_1,p_1q_1) $ and $ \displaystyle \frac{...
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Confidence Interval for Exponential

I am simulating exponentially distributed data with rate $5$ and I want to construct the confidence interval for usual convention $\alpha = 0.05$. ...
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central limit theorem in term of entropy

The usual central theorem uses iid samples. Are there a generalizing theorem to non-iid samples, using the conditional entropy of each sample given previous ones ?
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Central limit theory [closed]

I know that when you keep adding Sn it will tend to approximate a normal curve as n gets bigger and bigger,but what happens if you change the expectancy ?
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Posterior distribution of weight vector tends to Gaussian distribution as data size increases: is it true?

I'm working on Pattern Recognition and Machine Learning(Bishop), Chapter 6, which is about Gaussian Processes. Author says in page 315 : The usual justification for a Gaussian approximation to a ...
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Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread. Does this property have a name? Are there ...
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confusion about the link between residuals, error terms, sample size and CLT in ANOVA

I feel a little confused about the assumption of the ANOVA and what it ensures mathematically the errors have to be iid and normally distributed N(0,1). independance of observation. Is it not a ...
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Convergence in Distribution for i.i.d. data

Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
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Converges in distribution

Let, $X_1, X_2, \ldots, X_n$ be i.i.d. RVs with mean $0$ variance $1$ and finite fourth order raw moment. Find the limiting distribution of $Z_{n}=\frac{\sqrt{n}(X_{1}X_{2}+X_{3}X_{4}+\cdots+X_{2n-1}...
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Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
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Normal approximation to Poisson: With Continuity Correction the Approximation Seems Worse

This is Exercise 3 in Section 6.3 of Probability and Statistics, 4th edition, by DeGroot and Schervish: Suppose that the distribution of the number of defects on any given bolt of cloth is the ...
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When multiple realizations of uncorrelated but dependent random variables are added they become independent

I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance: $X \sim \mathcal{N(0,1)}$ $Y =...
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A failure of convergence of conditional distributions

Consider $n$ iid samples $(X_i,Y_i)$ generated such that $X_i$ is a truncated normal of mean $\mu_X$ truncated to the left of the origin, and $Y_i$ is a truncated normal of mean $\mu_Y$ truncated to ...