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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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Principle of all statistical tests [longread][discussion] [on hold]

I know that this is not the right type of question to be asked on stackexchange. If this question gets to much downvotes I will take it down. If somebody knows the appropriate place to discuss these ...
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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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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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1answer
61 views

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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1answer
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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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CLT for different but independent random variables [duplicate]

Is the Central Limit theorem still applicable if we consider a sum of independent but different random variables? (Each with finite mean and variance) Is there some theorems about this ?
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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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1answer
161 views

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 ...
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CLT for random sums: Anscombe's Theorem vs. “classical” version

Given a compound Poisson distribution $$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with $N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$ $X_{k}$ are non-negative iid random variables such ...
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1answer
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Are there any results on the distribution of the posterior mean across data sets?

I know that this question borderlines on Bayesian and frequentist philosophy, somewhat related to this question. Bayesian point estimation sometimes uses the mean of the posterior distribution. That ...
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Testing If Mal-Dosing Exceeds a Certain Threshold

Suppose you have mal-dosing data of some medication (in percent). You want to know if mal-dosing on average exceeds 5% regardless of over- or underdosing. Can I do a simple t-test? (if I assume iid ...
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1answer
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Help justifying a conditional variance step?

The problem runs: "The count of claims $N$ obeys a Poisson distribution with mean $\lambda$; the amount per claim $X$ obeys an exponential distribution of mean $\theta$. Let the variable $S$ be the ...
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What does “properly normalized” mean in CLT? [duplicate]

What does "properly normalized" mean in CLT? https://en.wikipedia.org/wiki/Central_limit_theorem In probability theory, the central limit theorem (CLT) establishes that, in some situations, when ...
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Standard deviation of a ratio and calculation of weight

I have several "skilled" and "unskilled" wage observations for a number of countries, and would like to construct a single skilled-unskilled wage premium by country as the ratio of the simple average ...
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Is a kurtosis of 1 unique to a equiprobable two-point distribution?

This question came up while having a look for a CLT for the sample variance which I found under this link (Theorem 5.3.2) from another stackexchange question which I can't remember. The Theorem from ...
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What are basic means for sample size determination?

I'm trying to answer experiment design questions of the following form: We take some measurements (samples) and want the sample mean to be $\pm \epsilon$ around the true value $\mu$ with 90% ...
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3answers
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Central limit theorem and residuals

I have often read that thanks to the CLT, the residuals of a model are asymptotically normal. This argument always seemed odd to me since CLT states that The sum of a number of independent and ...
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dividing sample variance with variance

Earlier today i came across something that i could not quite understand so i figured maybe someone here sits on enough knowlege to help me out. The basic question is that one should decide what $a_s$ ...
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Applying CLT to difference in sample proportions?

I am reading a derivation that has the following statement about a two-sample proportion test: $$ \frac{\hat p_1 - \hat p_2}{\sqrt{(\frac{1}{n_1}+\frac{1}{n_2})\hat p(1-\hat p)}} \stackrel{d}{\...
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Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
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Normal approximation on (what it looks like) a poisson

I am self-studying inferential statistics from Larson's introductory textbook named: "Introduction to Probability Theory and Statistical Inference" (1st edition John Wiley & Sons.). I came ...
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Difference in Outcome While Applying the Central Limit Theorem

I am currently writing some code to control a model train layout. As part of this, how much the distance travelled by a locomotive (under a constant speed) varies during a specific time interval is of ...
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1answer
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How to calculate difference between sample and population mean in R [closed]

I am taking a statistics course focused on R and was asked this question about a normally distributed sample of weights: Use the CLT to approximate the probability that our sample mean estimate is ...