# 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) ...
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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 ...
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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)...
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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 ...
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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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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