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Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "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)

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name for distribution of sample mean before standardization to t-distribution [closed]

I’m re-learning a very basic statistics of standard error of mean. When population variance is known, the distribution of sample mean is normal distribution, according to the central limit theorem. On ...
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Is Bootstrapping Independent Time Series to Construct Prediction Intervals Valid?

Question: I have a dataset consisting of multiple univariate time series, each representing an independent sequence of insurance claim amounts over time. My goal is to predict future claim amounts ...
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How do I calculate estimated variance for an ensemble forecast?

I have several (n) different forecasts of comparable quality for a variable, based on the same data but using wildly different statistical models. For each, I have generated an estimate for m periods ...
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Precision of estimates of lower bit error probabilities at higher SNR

For my university lab in wireless communications, I simulated a simple uncoded BPSK (binary phase shift keying) channel with AWGN (additive white gaussian noise) to estimate the BER (bit error rate) ...
Scarab's user avatar
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1 answer
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Can I add two independent results from the central limit theorems?

I'm reading introduction to mathematical statistics by R. Hogg, et al. I have some trouble to understand a limiting distribution. Let $X_1,\cdots,X_{n_1}$ be iid random variables from $Bernoulli(p_1)$ ...
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Does a random variable that includes the summation of independent samples from different distributions obey Central Limit Theorem? [duplicate]

I am learning from the book of statistics by sheldon M ross and it's a great book. However, I landed upon a small query that book failed to address me. According to CLT , sum of random variables when ...
CREATIVITY Unleashed's user avatar
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Central Limit Theorem to determine sample size

Given a sample $X_1, ..., X_n \sim^{iid} $ Bern(p). I want to test $H_0: p = 0.49$ vs. $H_1: p = 0.51$. How can I determine the sample size for which the probability of type I error (and type II error)...
JohnD's user avatar
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Construct transformations of random variables that are "more normal"

I am reading this page in the Encyclopedia of Mathematics about transformations of random variables. I am puzzled about the Example 2: Let $X_1,...,X_n,...$ be independent random variables, each ...
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Is this central limit theorem?

Context: I have an implementation of Wilson's algorithm for generating uniform spanning trees. After generating 1 million instances of USTs on a K5 (complete graph with 5 edges), I plot a histogram of ...
ayang's user avatar
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Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
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Is Central Limit Theorem about multiple samples or just one?

I've studied CLT and my understanding is that multiple samples will generate a normal distribution centered in the mean of the population. However, today, one post in Linkedin was saying that "...
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Weak Law of Large Numbers: Conditional Expectations in Random Subsequences

Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
Albert Paradek's user avatar
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Show convergence in distribution to a bivariate normal vector

Let $Y_1,...,Y_n$ be iid exponential random variables with mean $\theta>0$. Let $$ \hat\alpha_n:= \frac{1}{n} \sum_{i=1}^n Y_i \quad \text{and} \quad \hat\beta_n:=\sqrt{\frac{1}{n} \sum_{i=1}^n (...
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Why intuitively is standard deviation the correct thing to scale to get Central Limit Theorem?

Let me start off by saying I already know all the rigorous formulas, but let me explain why I still feel like something is missing in my understanding. There is no need for any answer going over e.g. ...
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Does central limit theorem apply to the following product case?

All! I know the central limit theorem works for a lot of cases, and some variants including Lindeberg are also pretty useful, but now I met such product case, and wondering if some classical or ...
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Possible reason for the Central Limit Theorem not to apply?

I have a multi-generator process which generates around 1200 data points per minute consisting of ~80 processes which contribute more or less evenly to the data points. Each singular process should in ...
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T Distribution and CLT

By definition, the T distribution is the ratio of standard normal variable and sqrt of scaled $\chi^2$ variable. The "popularized" version of (one sample) t statistic goes like this: $\frac{\...
Kaiwen Wang's user avatar
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Adapting two-sample $t$-test of a ratio for log transformation

I have some data, belonging to paired groups $A,B$. From each group I get a non-negative statistic $d^A,d^B$ which is averaged on all samples in group. My interest is in the ratio $\frac{d^A}{d^B}$, i....
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1 answer
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Non-standardized vs Standardized Data Normality

I have a dataset with over 900 cases. I performed a Q-Q Plot on the non-standardized HRV data (dependent variable), and the points formed a wavy curve indicating it does not follow a normal ...
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Asymptotic Normality for GEE Parameters

In the famous Liang and Zeger 1986 paper on GEEs https://www.jstor.org/stable/2336267?seq=9, they sketch a proof using the standard m-estimator arguments: (unstated) regularity conditions + first-...
Winston's user avatar
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Does rolling average help reduce generalize linear model on timeseries variable to OLS?

I have a data $Y_t$ that basically measures the number of certain events at any day, and I have a time-series for all countries from 2015 to now. And I am trying to fit that $Y_t$ to some $X_t$ that I ...
The One's user avatar
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3 votes
2 answers
401 views

How to use one-sample t-test on skewed distributions?

I have been reading that people use the one-sample t-test also for skewed underlying distributions, saying that for a high enough number of datapoints (I read for example N=30 and N=100 in some places)...
Mars's user avatar
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Convergence in probability and boundness in probability with respect to sample mean and sample variance

This is a question about the convergence in probability and boundness in probability. Suppose $X_i \overset{\textrm{i.i.d.}}{\sim} (\mu, \sigma^2 )$ for $i=1,2, \cdots, n$. Denote $\overline{X}$ and $\...
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2 answers
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Applying the Central Limit Theorem to a Piecewise PDF

For a large sample of size 1296, we are looking at independent and identically distributed random variables defined by the piecewise probability density function $f(x) = \frac{1}{3} (I_{(0,1)}(x) + I_{...
Occhima's user avatar
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1 answer
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Central limit theorem for independent but non-identically distributed random variables

My question is about proving the Lyapunov CLT (every mean is $0$, $\delta = 1$). It is similar to this question but without any assumption about the random variables following Bernoulli distributions. ...
johnsmith's user avatar
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Asymptotic normality of the maximum likelihood estimator with dependent data

In the setup, assume $\left(\mathbb{R}, \mathscr{B}\left(\mathbb{R}\right), P\right)$ is the underlying probability space and suppose that $\left\{\mathcal{F_n}\right\}_{n\in \mathbb{N}}$ is a ...
Yashaswi Mohanty's user avatar
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1 answer
42 views

Would the distribution of the average number of correct answers for a five-question game constitute an application of the central limit theorem?

Say you have a game in which users answer five questions daily. Would the distribution of average scores for a single user among all of their games meet the criterium for the central limit theorem? I ...
lp804's user avatar
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Normal Approximation in testing for association in a 2x2 contingency table

Suppose we have a 2x2 contingency table with binary RVs $X, Y$ and we want to test for independence. Let $n$ be the total amount of measurements. Let $n_{11}$ be the number of occurrences of both $X = ...
James's user avatar
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1 answer
81 views

Large sample distributions

Suppose we have observations $x_1, x_2, \ldots, x_n$ where $n$ is very large. Now we standardize the observations as $$y_i=\frac{x_i-\bar{x}}{\frac{s}{\sqrt{n}}},$$ where $s=\frac{\sum\limits_{i=1}^n(...
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2 votes
0 answers
54 views

Central Limit Theorem for t(2)-distributed random variables

Let $X_k \overset{iid}{\sim} t(2)$. Is there any limit theorem about $\bar{X}$? I know $\text{Var}(\bar{X})$ doesn't exist, so I cannot use classical CLT. But I believe there must be other means to ...
ChS's user avatar
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1 answer
143 views

Linear Regression - Proof that coefficients estimated via OLS follow a normal distribution [duplicate]

The aim of my question can be better illustrated by this quote extracted from the third chapter of Elements of Statistical Learning (link to book): I'm trying to understand why, given that the error ...
Frederico Portela's user avatar
2 votes
2 answers
109 views

Bayesian and frequentist connections regarding the central limit theorem

I have been wondering how the central limit theorem may be useful in Bayesian statistics with potentially misspecified model distribution. Suppose $x$ is a random variable that follows an unknown (and ...
fan455's user avatar
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3 votes
3 answers
283 views

How do we know the distribution of regression coefficients

I'm reading up on asymptotics and hypothesis testing and was thinking about how they link together with regression coefficients. I have read that the CLT shows that the standardised sample mean ...
Geoff's user avatar
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1 answer
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Sample size in Sample Proportions

I am in high school learning about sample proportions and they say that $n$ is the sample size. The example they gave is you spin a spinner board where the chance of landing on a 1 in 0.6, and 0 is 0....
Venkat's user avatar
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34 views

In a skewed sample with a large n, does Central Limit Theorem dictate that a t-test can be used, even if the mean cannot be interpreted? [duplicate]

I understand that, in the case of a highly skewed population and sample, the sampling distribution of the mean can still be normally distributed if the sample size is large, according to Central Limit ...
Josh Blake's user avatar
2 votes
3 answers
60 views

Testing the Anderson–Darling and Central Limit Theorem

I was trying to simulate the Central Limit Theorem in R. Unfortunately, even in large samples (e.g., 80), the Anderson–Darling test could not recognize normality. Therefore, I wrote the following code ...
Kώστας Κούδας's user avatar
1 vote
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37 views

Papers or documents about the central limit theorem and its possible extensions: what happen when the sample size is big? [closed]

The central limit theorem in its most popular form states that (without being too formal) for a set of random variables $X_1,X_2,...,X_n$ independent and identically distributed with mean $\mu$ and ...
lulufofo's user avatar
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1 answer
27 views

Distribution of the mean of samples taken from two different distributions

Consider you have some distributions $Z_1$ and $Z_2$ of which you select $n_1$ samples from $Z_1$ and $n_2$ samples from $Z_2$. We now add up these samples and take the mean. The question becomes, ...
wjmccann's user avatar
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0 answers
27 views

Are all examples of normal distribution in nature only a consequence of CLT? [duplicate]

I have read that many bell curves we see in nature is just a consequence of the CLT, because those things are just the result of many small additive causes (e.g. human height). Then my question is: ...
sitems's user avatar
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2 votes
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CTL vs normal distribition of residuals

Hello, I submitted a study with a linear regression model. My sample size is >1000, so I invoked CTL. The statistics reviewer is requesting that we check assumptions of normal distribution (...
Alstor's user avatar
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7 votes
1 answer
369 views

What is the probability space that the CLT is really being applied to?

Can someone please walk me through (or cite a reference to) exactly which fixed probability space is used in an application of the CLT, especially in the finite probability space case? My question ...
ac1501's user avatar
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3 votes
2 answers
98 views

What is the rigorous justification for applying LLN or CLT to finite probability spaces?

Both CLT and LLN are stated in terms of a fixed probability space that admits an infinite sequence of IID RVs. It is a common-place in many probability and statistics texts/notes that such a sequence ...
ac1501's user avatar
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2 votes
0 answers
79 views

How to prove that if the bootstrapped sample means are approximately normal, then the sample mean is approximately normal?

I'm considering the scenario described in this question, namely: we have IID $X_i, i = 1, ..., n$ sampled from a population with mean $\mu$, variance $\sigma^2$ unknown the goal is to put a ...
travelingbones's user avatar
1 vote
0 answers
51 views

Why does the normal distribution show up everywhere? [closed]

I know that the common explanation for this is the Central Limit Theorem being applicable to most probability distributions. However, I can't seem to apply the Central Limit Theorem to simple examples ...
timeinbaku's user avatar
1 vote
0 answers
17 views

Could the numerical value of mean square error (or root mean square error) tell us something about the rate of convergence?

Suppose I have an estimator $\widehat{\theta}$ for $\theta_0$ that is root-n consistent and asymptotically normal. In the monte carlo simulations of many papers that I've read, consistency is usually ...
ExcitedSnail's user avatar
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3 votes
1 answer
165 views

Does the Central Limit Theorem Apply to All Finite Samples Even If They Come From Distributions That Don't Have a Finite Variance?

Some distributions, like the Cauchy distribution, don't have a finite variance, and therefore the central limit theorem does not apply to them. If I have a thousand randomly selected observations from ...
David Moore's user avatar
2 votes
1 answer
74 views

Techniques/diagnostics for gaining confidence in normality assumptions and resulting confidence intervals

I have data that is reasonably assumed to be iid samples from some distribution. Our goal is to put a confidence interval on the population mean and have something similar for the population variance. ...
travelingbones's user avatar
6 votes
3 answers
558 views

Why don't we use normal distribution in every problem? [closed]

I was reading about normal distributions and the Central Limit Theorem (CLT) and I came up with a question. Why do we bother ourselves to use machine learning techniques when the CLT gives us the ...
George Wilhelm Hegel's user avatar
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1 answer
43 views

Interpreting the Concept of 'Single Sample Normality' in the Context of the Central Limit Theorem

In the context of the Central Limit Theorem (CLT), which postulates that the distribution of sample means will approximate a normal distribution given a sufficiently large number of samples and sample ...
Amit S's user avatar
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245 views

Monte Carlo simulations and Central Limit theorem

I am simulating the revenues of a portfolio of items using one input variable. This variable is randomly extracted from a normal distribution n times, where n is the number of Monte Carlo simulations. ...
floyd123's user avatar

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