# 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 ...
1 vote
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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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1 vote
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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 ...
1 vote
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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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### 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}(...
1 vote
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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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### 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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1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
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=...