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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7answers
13k views

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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2answers
160 views

Central Limit Theorem - Vector of Random Variables - Imputation of Missing Values

I have a large dataset -- 300,000 records, each representing a customer-- and a variable holding their incomes. Since there were missing values, I used the Multiple Imputation Chained-Equations ...
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1answer
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Implications of zero limiting variance

Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$. Can I claim that for large enough $n$ ...
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1answer
28 views

confidence interval for sample that is not normal distributed

I have one sample of only 86 values, which is not normally distributed according to the Shapiro-Wilk normality test. Can I still use this formula/code (sorry R code) to estimate the 95% confidence ...
2
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0answers
48 views

Confidence Interval for Estimator using Delta method

The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
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2answers
5k 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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1answer
98 views

theoretical confidence interval depending on sample size [closed]

I am using R and plain English to express my question. Let us say I have a "true"/made up population, which is normally distributed with a mean of 500000 and a standard deviation of 13000: <...
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1answer
46 views

Does Normality of a Time Series imply Stationarity and Viceversa?

I have a theory question which never became completely clear to me. Reading Hamilton (1995) I understod that the stationarity requirement for time series data stands as the normality requirement for ...
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1answer
43 views

possible use case of central limit theorem for analysts

This is a bit of a long shot but I would appreciate any help please. I have to do a basic stats course for our analysts, which I try to make as applicable and useful as possible using our data (e.g. ...
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2answers
42 views

Does the Central Limit Theorem imply that $(\hat{X}_n - \bar{x}) = o_p(1)$ at rate $O_p(1/\sqrt{n})$?

Let $\left\{\hat{X}_n\right\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, i.e., $\left(\hat{X}_n - \bar{x}\right) = o_p(1)$. Then say that, by some applicable ...
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0answers
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ICA: a question about the non-gaussian requirement

I'm new in the ICA processing and I'm trying to understand the non-gaussian requirement. I read that the problem is that, if the composed data is $\mathbf{x}=\mathbf{As}$ with $\mathbf{A}$ (unknown) ...
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1answer
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Rate of convergence of $\hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1}$ to the probability limit?

Consider the simple linear regression model. $$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n. $$ Let $\mu_x$ and $\sigma_x^2$ represent the mean and variance of ...
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0answers
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CLT-based confidence intervals not working in code

I wrote code to draw $n$ samples from a categorically distributed random variable $C$ with probabilities $p_i$ for each value $i$ and to use those samples to compute an approximation $q_i = n_i/n$ of ...
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0answers
22 views

Understanding rate of convergence for realized estimators

I'm a Econ student currently taking a small course on realized measures/estimators. I'm a bit confused about the meaning behind rate of convergence for each type of estimator. I'll give some ...
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2answers
71 views

Central Limit Theorem and Skewed Distribution

I'm looking for a simple answer to this question relating the central limit theorem and Gaussian and skewed distributions, if one exists. I used the binomial function to generate calculations of the ...
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2answers
49 views

Does a vast amount of probability and statistical literature make a mistake when they make use of CLT/asymptotic normality?

Suppose we toss a fair coin $N$ times and we are interested in the probability that we get at least $cN$ heads for $c\in [0,1]$. We can model this situation by letting $S_N = \sum_{i=1}^N X_i$ where $...
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1answer
41 views

Central Limit Theorem for Truncated observations

Consider a random variable $X$ with values in $\left[0,\infty\right)$ such that $E\left[X\right]=\infty$. Given $M > 0$ I want to estimate the expected value of $X$ truncated at $M$. That is I want ...
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1answer
60 views

variance of difference of two means $\bar X-\bar Y$

For the mean of sample $\bar X$, $\frac {\bar X -\mu}{\sigma_X/\sqrt{N}}$ has normal distribution. According to CLT, $\bar X$ has a variance of $\sigma_x/\sqrt{N}$. For two means $\bar X, \bar Y$ of ...
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0answers
15 views

Whats the difference between sample size and sampling size?

I am currently studying a r code that's used for verifying the CLT and in that function, there are three inputs. First is n which is the number of observations second is a parameter for a geometric ...
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1answer
41 views

How to use CLT on statistical inference?

I have an issue for how a sample of the population can be used to infer about the population parameters. For example, see the following questions: The average weekly earnings for female social ...
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0answers
17 views

Finding the p-values for one sample t-test in different cases

Calculate and interpret the $p-$ value in the following situations: (a) one-sample $t$ - test for testing (i) $H_{0}: \mu=\mu_{0}$ vs $H_{1}: \mu \neq \mu_{0}$ (ii) $H_{0}: \mu \leq \mu_{0}$ vs $H_{1}:...
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0answers
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Central Limit Theorem - does the number of times the samples are taken matter in terms of the CLT?

Let's say that in R, we generate $n$ random variables $Y_1, \dots, Y_n$ which all follow an exponential distribution. We then construct the mean $\bar{Y_n}$. The process is repeated $m$ times, so we ...
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1answer
45 views

Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller?

When you add two independent normal distributions the resulting distributions' variance is the sum of the variances i.e. it gets larger. However, the Central Limit Theorem states that when ...
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1answer
46 views

Central Limit Theorem formula transformation with iid variables

I was looking into Central Limit Theorems and how a CLT is derived and I found this source quite helpful. The only thing I am having trouble to comprehend is the transformation of the formula $$\frac{\...
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0answers
30 views

Sampling distribution of Pearson correlation coefficient

Suppose I have draw a random sample of points $(x_i,y_i)$ iid from some distribution, then I compute the Pearson correlation coefficient $\overline{\rho}$ of the points in the sample. Is $\overline{\...
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1answer
37 views

Do I need to sample the distribution in order to apply the central limit theorem?

My scenario is the following: I have a sample set of 1000+ rows, of which my variable of interest is a non-normal distribution. I haven't tested throughtly to check what kind of distribution it is, ...
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1answer
52 views

Central Limit Theorem and Normal Distribution

I have the following questions as homework where I have to decide whether they are true or false. a) The standard deviation of the distribution of the sum of independent random variables is also ...
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0answers
18 views

Convergence of Normalised Sum of IID Random Variables

I have a Markov chain X that starts from the stationary distribution. Let define $S_n = X_1 + \cdots + X_n.$, where $X_i$ is the state of the Markov chain. Let's have 3 states. I wanted to prove the ...
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2answers
40 views

How to interpret confidence interval and prediction interval in simple regression “in/with the context of sampling distribution”?

With the context of sampling distribution, in regression analysis, is the following an appropriate interpretation? Assumptions : X & Y have a linear relationship sample size is large enough for ...
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0answers
19 views

How is the Central Limit Theorem related to Maximum Likelihood Estimation?

While studying the Maximum Likelihood Estimation, I often hear that the Central Limit Theorem kicks in do to the Confidence Interval in relation to the Maximum Likelihood Estimation. As far as I know, ...
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0answers
118 views

Why does MLE tend to normal distribution

We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$). If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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1answer
62 views

Convergence of random walk in $R^2$ to the Brownian motion on circle

We know that the random walk generated in $R^1$ can converge weakly in distribution to the Brownian motion in $R^1$. Could anybody provide a mathematical proof, how a random walk generated in $R^2$ ...
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0answers
19 views

Is there a law or concept related to unchanging average?

I have a survey comprising 30 questions with 50k respondents where respondents mark each answer on a 5 point scale. An average of all the questions for all the respondents is metric we track monthly. ...
2
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1answer
115 views

Confidence interval true/false questions

A sprinkler system is being installed in a large office complex. Based on a series of test runs, a 99% confidence interval for the population mean $\mu$, the average activation time of the sprinkler ...
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3answers
22k 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 ...
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0answers
21 views

Weak Law of Large Number to Central Limit Theorem [closed]

Consider a distribution with unknown mean μ and population standard deviation σ=30. Using the Weak Law of Large Numbers, what is the minimum sample size in order to attain a probability of at least 99%...
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0answers
13 views

Confidence interval for average of different binomial trials - can I apply central limit theorem?

I have person-level data, and for each person, I have the racial breakdown of the geographic area in which they live. For example, person A might have 100 people in their area, and 80 (80%) of them ...
2
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1answer
134 views

Proof of multivariate central limit theorem

$\newcommand{\phi}{\varphi}$ $\newcommand{\eps}{\epsilon}$ I'm using the book called 'A Course in Large Sample Theory' from Thomas S. Ferguson. During studying the proof of the central limit theory in ...
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0answers
33 views

Convergence in distribution and Slutsky's theorem

It is known that from the CLT, if $X_i \stackrel{\text{iid}}{\sim} F$ for some distribution $F$ with finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\...
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1answer
66 views

How is confidence interval related to central limit theorem?

I am currently looking into Confidence Interval and sees that Confidence Interval is possible based on Central Limit Theorem. So, I have been looking for how Central Limit Theorem is related to ...
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0answers
64 views

Interpretation of odd Central Limit Theorem (i.i.d) condition

My class was taught a third sufficient condition for the CLT to hold in the i.i.d. case that can replace the Lindeberg or Lyapunov conditions. I have never seen this condition before and am wondering ...
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0answers
45 views

Asymptotic Distribution Using CLT

I have random variables $X_1, X_2, ... , X_n \sim \text{IID } f_X$ using the density function: $$f_X(x) = \frac{2x}{\theta^2} \cdot \mathbb{I}(0 \leqslant x \leqslant \theta).$$ I have to use the ...
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1answer
142 views

Does the sum of discrete uniforms coverge to a discrete Gaussian?

Is there some analogous of the Central limit theorem for discrete uniforms and discrete normal distributions? To be more specific, let's say we have identical and independent random random variables $...
39
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6answers
3k views

Debunking wrong CLT statement

The central limit theorem (CLT) gives some nice properties about converging to a normal distribution. Prior to studying statistics formally, I was under the extremely wrong impression that the CLT ...
3
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1answer
45 views

Central limit theorem - num random variables vs. sample size?

Does the Central Limit Theorem require the number of random variables to increase to a sufficiently large number or the number of samples of each random variable to increase to a sufficiently large ...
3
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0answers
196 views

Central Limit Theorem - interval estimation

I'm rolling a regular dodecahedron (12-side die) 1200 times. I need to find an interval, in which the total count of prime-number results will lie with the probability of 95%. I have to use the ...
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0answers
41 views

Can I usefully apply the Lyapunov CLT condition to a finite sum of Bernoulli random variables?

I'd like to get a CLT-like approximate distribution (mostly tail behavior) of the sum $X$ of $n$ independent Bernoulli random variables $X_1, \dots, X_n$, with proportions $p_1, \dots, p_n$. The ...
2
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0answers
35 views

CLT - Adding small samples to one big sample

According to CLT, the SE is the SD of the distribution of several samples means. This SE depends on each sample mean, the SD of each sample and N (the size of each sample which I test). Since there ...
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0answers
52 views

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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0answers
24 views

Asymptotic distribution of mme of coefficient of variation

For random sample from unif(0,1) distribution, method of moments estimator for coefficient of variation is sample mean divided by sample standard deviation. Here, coefficient of variation theta is mu/...

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