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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Convergence of standarized empirical distribution functions to distribution of standarized sample mean

Suppose we have a family of $i.i.d.$ random variables $\{X_n\}_{n=1}^\infty$, with distribution function $F(\cdot;\mu,\sigma^2)$, with mean $\mu\in\mathbb{R}$ and variance $\sigma^2>0$. By virtue ...
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CLT for t-test with unequal sample size; one group < 30?

CrossValidated has many discussions on how unequal variances are not a practical issue for two-sample t-tests when Welch correction is used and on how normality assumptions do not play a role (in Type-...
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Variance and Mean Relationship from Simulated Poisson Process with Sampling

Background: I have a simple simulation of a sampled thresholded Poisson Process that arrives at a closed form solution but need help with the proof. In my example, I am simulating 10,000 silicon ...
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When to model assuming a Poisson vs when to model assuming Normal?

I understand that If I add an infinite number of variables the limiting standardised distribution is standard normal. This comes from CLT I also understand that the summation of independent Poissons ...
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Central limit theorem with sorted samples

I recently encountered a strange situation while dealing with sampling : Let's suppose I have X1 ... Xn random samples drawn from a population. Then I sort every samples and I make the sum of every ...
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I learned that CLT is about unbiased statistic, like means. What about sum of distributions (not sampling sums)?

Let's assume I sample many means from any distribution (that has the 1 raw moment). It should resemble the normal distribution. I heard that the same works for medians, variance, range and any other ...
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Show that for $\alpha\ge-\frac{1}{2}$, the central limit theorem holds

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables such that $$P(X_n=n^{\alpha})=\frac{1}{2}=P(X_n=-n^\alpha),$$ for some $\alpha\in\mathbb{R}$. Show that for $\alpha\ge-\frac{...
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Events per time given time that an event takes to happen

In a book I read this affirmation: "Hard disks are reported as having a mean time to failure (MTTF) of about 10 to 50 years [5, 6]. Thus, on a storage cluster with 10,000 disks, we should expect ...
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Why should i care about independent triangular arrays?

In Advanced probability class we were building up to proofs of the central limit theorem. We started by stating the theorem of the Classic CLT, then we introduced the independent triangular array and ...
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root n consistency of parameter in mixture distribution

We have iid observations $\{X_i\}_{i=1}^n$ from CDF $\theta G + (1- \theta)H$ where $\theta \in (0,1)$ is unknown. Find a $\sqrt{n}$ consistent estimator for $\theta$ using the observations. Note that ...
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Central Limit Theorem with Exponential and Uniform Distributions

The waiting time of a customer in a customer service telephone line in company number 1 has the exponential distribution with an expected value of 2.2 minutes. The waiting time of company 2 has the ...
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Is there a statistic such that for large sample sizes $a_n (\hat{\theta} - \theta) \sim N(0, \Sigma)$ approximately but $a_n \neq n^{1/2}$?

Various central limit theorems are of the form $a_n(\hat{\theta}-\theta)\sim N(0, \Sigma)$ approximately as $n \to \infty$ and usually $a_n = n^{1/2}$. Are there central limit theorems for statistics ...
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Multivariate Lyapunov CLT?

The univariate version of the Lyapunov CLT (a version of the CLT for independent but not necessarily iid random variables) is as follows according to Wikipedia (https://en.wikipedia.org/wiki/...
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two sample t-test for non-normal population

I assume this question has been beaten to death and thus I am just looking for a reference which goes through the details. Assuming all populations we deal with have finite means and variances (even ...
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Test for equal parameters of two regression models: compare coefficients directly or check if interactions are zero?

I have two data sets and obtain regression models with coefficient vectors $\beta_1$ and $\beta_2$. I want to test $$H_0: \beta_1 = \beta_2$$ against the alternative that the two vectors are not equal....
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Correcting for systematic bias in test size

Let $X_1,\dots,X_n$ be iid random variables with mean $0$ and variance $\sigma^2$, and let $$\xi:=\frac{1}{\sqrt{n}}\sum_{i=1}^n (X^2_i-\sigma^2).$$ Then, by the standard CLT, we have $\xi\Rightarrow ...
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How to simulate non-gaussian stochastic paths

(Edited to be clearer) I am trying to replicate simulating Geometric Brownian Motion (GBM) but instead of the stochastic increment following a normal distribution, I would like it to follow a ...
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Obtaining formulae for Poisson confidence interval

I have a sample $X=(X_1,\dots,X_n)$ of $i.i.d$ Poisson variables such that $n=100,\overline{X}=8.8$. My goal is to obtain a $80\%$ confidence interval for the parameter $\lambda=\theta$. That is, the ...
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a good estimator in 2-stages least-squares

I am now studying the 2-stages least-squares method and have been curious about the following circumstances. Suppose that I have $Y_i = X^{T}_{i}β +e_{i}$ with $\mathbb{E}(e_{i}X_{i}) ̸\ne 0$, that ...
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Why can de-noising diffusion models be sampled with Gaussian distributions?

In de-noising diffusion models 1 the latent is typically sampled with a unit normal distribution, and then the sample (e.g. image) is generated by iteratively removing noise during the backwards ...
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Using CLT to get Confidence Interval for Mean in MA(1)

Self-study: Let $$X_t = \mu + a_t + \theta a_{t-1}$$ $a_t$ is white noise with mean 0 and variance $\sigma^2$. Given $\bar x = c$ for a sample size of 100. Find confidence interval for $\mu$. My ...
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Conditions to apply Lyapunov's Central Limit Theorem

Let $\left\{X_{1}, X_{2}, \ldots X_{k}\right\}$ denote a set of $k$ IID $\operatorname{Bern}(p)$ random variables. Also, I have a set of $k$ non-negative integer weights denoted by $\left\{a_{1}, a_{2}...
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How to find 95% CI of CI of Mean?

We know that the 95% confidence interval around the population mean is calculated as x̄ ± 1.96* σ/√n, where x̄ is the sample mean, σ is the sample standard deviation and n is the sample size. On the ...
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Sum of 5 normal distributions [closed]

what would be a correct solution to following problem: The medical costs per patient are normally distributed with a mean equal to 7 euro and a variance equal to 49. Compute the probability that the ...
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Are there distributions for skewness and kurtosis? Similarly to mean (normal) and variance (chi-squared)

My question is really straightforward. The distribution of the sample means approaches a normal distribution (CLT). The distribution of the sample variance approaches a chi-square distribution (...
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understanding the Central Limit Theorem with different sample size

I'm trying to understanding the standard deviation of the sampling distribution from the Central Limit Theorem. $\bar{X}\rightarrow (\mu ,\frac{\sigma^{2} }{n})$ I can understand from a mathematical ...
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A problem in convergence and limit

For any non-negative integer $n$ and some finite $r$, we introduce the notation $n_k$ which indicates the number of $\{X_1, X_2, \cdots , X_n\}$ belonging to the $k$-th distribution type, for $k=1, 2, ...
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Converge in distribution and Big O pe

I have read an econometrics textbook on "converge in distribution" and found the following sentence. \begin{equation} \text{If} \hspace{0.25 cm} Z_{N} \xrightarrow{d} Z, Z_{N} = \large{O_p}(...
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A new convergence problem for the conditional expectation

You have risks $X_1$, $X_2$, ... (they are assumed to be independent, but not necessarily identically distributed) and $S_n= X_1 + X_2 + \cdots +X_n$ QUESTION: under what reasonable conditions do we ...
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Derive the variance of the standardized normal of sample mean

I have solved the following problem and felt a little bit uncertain about the my answer. Here is the problem. Let $Y_i \in L_{2}$ for $i=1,2,...,N$ be a scalar random variable with iid with $\mu_Y=E(...
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if I have a variable in my data that is not normally distributed, could I still use an ANOVA if my sample size is large enough?

I have a sample size that is more than 30 but one of my continuous variables is not normally distributed. The Shapiro-Willk test does say that my data is not normally distributed. I am going to ...
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When is the z-score/wald-score results will be a non-normal distribution?

If the hypothesis is valid the distribution of scores from several samples from the same population will follow a normal distribution for the z-test. That is true? I've made a simple experiment to see ...
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Central Limit Theorem for proportion - np >=10 or 5?

While trying to understand deeper about Central Limit Theorem for proportions, I learned CLT for proportions is based on the fact that binomial distributions can be approximated by normal ...
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Testing whether variables are identically and independently distributed

I have the probabilities of variables that we assume initially are independently and identically distributed random variables. I know that in actuality they are not from my knowledge of the problem, ...
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Random samples within the Central Limit Theorem - why select with permutation with repetition?

"The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the ...
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Sampling from an infinite population

SAMPLE SIZE I am sampling astronomical data and measuring clusters of galaxies (within a range of 100 megaprsecs) as nodes in a complex network, where k is a metric ...
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Optimum sample size for a large population?

I am trying to identify what should be the optimum sample for a population. Based on various articles. I identified that there is a formula that helps to calculate that , based on Confidence Level, ...
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What distribution for i.i.d. $X_i$ satisfies $\frac{\sum_i ^N X_i }{\sqrt{N}}=O(1)$?

What kind of statistical properties does $X$ need to have in order to satisfy $\frac{\sum_i ^N X_i }{\sqrt{N}}=O(1)$ where $X_i$'s are i.i.d.? Zero-mean?
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The sum of $n$ zero-mean values is of $O(\sqrt{n})$ at max

Why is the sum of $n$ zero-mean values is of $O(\sqrt{n})$ at max? If $X$ is a random variable with a standard deviation $\sigma$, the standard deviation of $n$ such i.i.d. random variables is $\sqrt{...
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Do you ever test the pre-asymptotic behaviour of a variable before performing a comparison of means t-test?

I understand that it is not the normality of a random variable that matters in a t-test, but rather the fact that the distribution of the mean follows a normal distribution for large samples. However, ...
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Asymptotic MLE Distribution With Two Random Samples

I'm studiyng for an exam, and I found this problem which I can not managed to solve... I will be really grateful if someone can help me, thanks you. Let $\left\{X_{1}, \ldots, X_{n}\right\} \sim^{...
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Central limit theorem for dependent binary-related variable

Let $Y\sim N(\mu, \sigma^2)$ and given sample size $n$, we have an iid sample $\{Y_1, ..., Y_n\}$. We sample $X$ (size $n$) from Bernoulli with probability $\pi$. Denote $Z_i=X_iY_i$. Then, when $X_i=...
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(Sums of dependent random variables) This problems develops a central limit theorem for a sum of dependent random variables

Let $X_1, X_2,...$ be i.i.d. r.v.s with zero mean and unit variance. Define $Z_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n X_jX_{j+1}$. (a) Show $Var(Z_n) = 1$ (b) Show $Z_n \to \mathcal{N}(0,1)$. Hint: First ...
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Using the Delta Method to get confidence intervals for a function of a parameter

I'm trying to use the Delta Method to get a confidence interval on some function of a population parameter $\theta$. Suppose we want a 95% confidence interval on $\frac{1}{\theta}$, where we're given ...
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What's wrong with this interpretation of a 95% confidence interval?

Note: I asked a version of this as part of another question, but I'm re-asking it as a stand-alone question with more detail. I've been trying to come up with more intuitive/less confusing ways to ...
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1 vote
2 answers
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Trouble understanding the Central Limit Theorem's real life application

As I understand it, the CLT states that for a sufficiently large sample size "n", the sampling distribution of the mean from a given population will approximate a normal distribution. Also ...
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how to prove if central limit theorem would apply to a process?

I'm very new to stats so apologies in advance for not using the right language and terms. I have just learned about the central limits thm can show up in places other than taking a sample distribution ...
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Central Limit Theorem and the Sample Sum

This 27.1 The Theorem lists two equations for Z: $$ Z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt n}}\\ Z = \frac{\sum_{i=1}^{n} X_i-n\mu}{\sqrt{n}\sigma} $$ Is it correct to say that the second equation ...
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2 votes
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Estimate population mean from sample with known distribution

My sample size is around 100k - should be enough for CLT to work. One intuitive approach is to estimate population mean by sample mean y. However, after plotting the sample data, I notice the ...
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Central Limit Theorem Practically Speaking

Suppose that $s=\{a, b\}$ where $a$ is the event that Tom takes a pill on a single day and $b$ is the event that Tom doesn't take a pill on a single day. Let $$ X(x)=\left\{\begin{array}{ll} 1 & x=...
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