# 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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### Steps in CLT proof unclear

In john rice Mathematical Statistics and Data Analysis we find a proof about the central limit theorem. Let $X_{1} , X_{2}...$ be a sequence of independent random variables having variance $\sigma^{2}$...
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### Central Limit Theorem - intuitive explanation without deep math [duplicate]

The Central Limit Theorem says that the distribution of the sample mean is approximately normal. Is there any intuitive explanation for why this should be so? I know it can be proven with deep math, ...
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### Where does "Uniform" in "Uniform central limit theorem" come from?

We may all know about the CLT. Today I have seen two articles where the use a new term (to me), that is "Uniform central limit theorem". A uniform central limit theorem and efficiency for ...
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### Central limit theorem and strong law of large numbers

I had a question in my mind , if a i.i.d distribution function follows central limit theorem , does that mean it will follow Strong law of large numbers also ?? Since in both cases sample means tends ...
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### $Y$ is equal to the sum of $n$ independent identically distributed Gaussian distribution variables, where $n$ is Poisson distribution

$Y$ is equal to the sum of $n$ independent identically distributed Gaussian distribution variables, where $n$ is Poisson distribution. If $Y$ is approximated to Gaussian distribution, what is its ...
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### Why do we apply the sample mean version of the CLT for a problem involving a sample size of 1?

I am having problems understanding the following question and answer. It seems to me that the sample size is n = 1 and the population size is N=500. If I read it this way then we do not have a large ...
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### How to apply Lyapunov CLT to data

I have a situation where I have around 30 classes of variables with different means and variances (though the means aren't too far from eachother; think 4-7) and that the distributions are right ...
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### Issues with sampling distribution over bootstrapped monte carlo simulations

Facebook posed an interview question (see ~49 min mark), how many days would it take (in days) to sample every user from a population of 1000, given that you sample 10 users/day each day? Analytically,...
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### Intuition behind $2\Phi(x)-1$

When using the central limit theorem to calculate for $n$ I come across having to use $2\Phi(x)-1$ to find this. However, I'm unsure why I have to use this and what it means and how it's derived. It ...
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### Testing the Central Limit Theorem with the Shapiro-Wilk test on dice rolling simulations

Rolling a single dice repeatedly will result in a uniform distribution. But if we roll multiple dice the sum would be the Normal distribution due the Central Limit Theorem (CLT). To verify this, ...
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### The one-sample t-Test/z-test and inference validity

The CLT tells us that as we collect the means of different samples, the sampling distribution resembles a normal distribution and this way we can infer with a CI on the sampling distribution, the ...
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### CLT application to exponential distribution? [duplicate]

I'm a little confused how the CLT can apply to aggregations of the exponential distribution. It's my understanding that the CLT says, in plain English, "sample means from virtually any ...
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### Central Limit Theorem with Bounded Sum of Variances?

I have a sequence of bounded independent random variables $X_1,...,X_n,...$ satisfying $\sum_{i=1}^{\infty} \mathbb{E}[X_i] < \infty$, $\sum_{i=1}^{\infty} Var[X_i] < \infty$. Most versions of ...
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### Mean Accuracy and Standard Error of the Accuracy for KNN Classification algorithm

The given below code snippet is from the assignment of online course IBM ML with Python. Here's the assignment. The used variable names :mean_acc and ...
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### Central Limit Theorem: Is the likelihood of obtaining some sample mean exact when n is not infinity?

The central limit theorem states: $$\lim _{n\to \infty}{\sqrt{n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)} \sim \mathcal{N}(0,1)$$ Which means if I ran an infinite number of experiments, ...
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### If $X_1, ..., X_n$ come from $\exp(5)$ using CLT, Calculate the probability $P(Z<z)$ [closed]

Please advise if the approach i am taking below is correct, I don't see how else to go about solving this problem. ...
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### What, if any, asymptotic arguments are used in moving between various statements of the central limit theorem?

What, if any, asymptotic arguments are used in moving between the various statements of the central limit theorem (e.g. in terms of sample means compared with standardised sample means)? Context. My ...
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### How to know the number of subjects and events required for statistical significance? [duplicate]

Say I conduct a study where i measure physiological signal of a set of N subjects, to record a particular event. I am interested of the duration and the amplitude of that event. The study ends when I ...
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### How can the CLT fix OLS regression residuals that are not normally distributed?

I often hear that when the residuals depart from normality, the central limit theorem can be used to fix things. I do not quite understand how this works, since the central limit theorem is a ...
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### Central limit theorem in practice and Berry-Essen

In a first course to statistics, confidence interval calculation using the Central Limit Theorem is introduced. Are confidence intervals sensible in practice? That is, given a 99% confidence interval ...
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### why n>=30 for central limit theorem to hold? [duplicate]

From population choosing samples(size n=30) and calculate its mean then repeating it N times will converge to normal distribution as N->inf when mean of each sample is plotted as a histogram. From ...
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### How to understand taking a simple sample to compute the Confidence Interval, using CLT?

Suppose there is a population distribution having mean M and standard deviation SD. We want to estimate the Confidence Interval by using a single sample using CLT. We take a sample and compute it's ...
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### Non-normal sample from a non-normal population (option returns) does the central limit theorem hold?

I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the ...
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I am looking for a formal proof that, with the CLT transformation, a random variable $Y \sim POI(\lambda)$ converges to a normal distribution ($Z\sim N(0,1)$). I believe this can be formulated as: $$... 0answers 8 views ### Original vs Sample mean distribution in reconstructing time series data Lets assume that the time series corresponds to some observable signal from a processor that computes a modulo 2 operation. The nature of the time series depends largely on the output of the modulo ... 1answer 49 views ### What is the meaning behind a sample distribution if you only have one sample? I'm trying to understand the meaning behind the central limit theory and the importance of CLT for inferential statistics. The problem that I encountered has to do with sample distributions. ​ I do ... 2answers 28 views ### Sampling as adding random variables, especially binomial RVs Is sampling equivalent to adding random variables? I'm a bit confused because as we can see that the binomial distribution becomes more and more shaped like a normal distribution as n increases. We'... 0answers 29 views ### PDF of given complex exponential equation of random variables [closed] Let d_i and d_k both are independent and identical random varibles with pdf \frac{3d_i^2}{R_{max}^3-R_{min}^3} where R_{max} and R_{min} both are constants and  R_{min}<d_i<=R_{max}... 0answers 60 views ### CLT in Diebold & Mariano (1995) The Diebold-Mariano (DM) statistic is derived as follows: Assuming the loss-differential between the two models d_t is covariance stationary:$$\begin{cases} \mathbb{E}[d_t] = \mu> 0 & \...
Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...