# 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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11 views

### Is this proof of convergence in probability to zero correct?

I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
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### Why are my sampled values are non Gaussian?

I just have a quick question regarding Importance Sampling Monte Carlo integration. If I sample from some pdf, $p(x,y)$, to calculate an integral. I.e., $I = \int f(x,y) \ dx\ dy$ It can be ...
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### Can the t-test be used to test the difference between percentiles of 2 samples? [duplicate]

For example, instead of testing the difference in mean, I want to test the difference in the 75th percentile of the 2 groups. Does the central limit theorem hold? and what would be the equation for ...
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### Why does the Normal Distribution have inflection points at +-1 standard deviations?

Supposedly this was Laplace's first error curve: Small errors occur more frequently, large errors less frequently; the shape of the Laplace Error Curve above roughly makes sense to me. Looking at ...
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### How to derive this MAE error bound on the central limit theorem?

Is this derived from Chebyshev's inequality or a tail bound theorem? If not, how was it derived? Does this require the existence of the third moment? Does this bound suggest the normal approximation ...
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### Central limit theorem seems counterintuitive given Law of large number

From what I understand, the Central limit theorem says the sample mean is distributed normally when sample number tends to infinity. However, the Law of large number says sample mean converges in ...
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### Asking for feedback on the application of a Central Limit Theorem

Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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### What's the relationship between $\frac{1}{n}\sum_{i=1}f(X_{1i},X_{2i})$ and $\frac{1}{n^2}\sum_{i}\sum_{j}f(X_{1i},X_{2j})$?

Suppose $(X_1,X_2)$ is a bivariate random vector following distribution $G$. $f(x_1,x_2)$ is a known bivariate smooth function. Suppose we are interested in estimating $E[f(X_1,X_2)]$ using a random ...
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### uncorrelated signal and noise

I referred a research paper in which it is written that " as the signal and noise are uncorrelated , so with increase in number of samples the product of signal and noise divided by number of samples ...
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### Pearson residues applied to the binary model

My question would be about Pearson residuals applied to the binary model. When we build confidence intervals for a proportion, p: we have a sample, and we count the number of individuals who have a ...
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### How is the normality of error term related to the standard error that is computed by statistical software for a particular coefficient of a variable?

1)My instructor says that because we assume the normality of the errors, we can calculate the correct standard error for the coefficient of a variable and further their t-statistics and p-values, but ...
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### Is CLT and AB test valid , when the test have repeat visitors making repeat pruchases

The Central limit theorem states that variables in population of the mean should be independent. But, when doing an AB test in a website, we have large numbers of visitors that will be coming back to ...
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### CLT + Slutsky for the t-test

If we want to test a mean and are lucky enough to know the populatio variance, we can use a z-test. Even if our population is not normal, for a sufficiently large sample, we can appeal to the central ...
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### How could it be that a skewed distribution produces a small standard error (smaller than the standard error taken from two normal distributions)?

I have three variables. Both Var1 and Var3 have approximately normal distributions, but Var2 has a right-skewed distribution (still fairly normal though). What I did: I took 100 SRSWOR (n=250) from ...
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### Are there situations where generalized CLT cannot hold?

Back in my university I was taught that gCLT and stable distributions are the solution for the infinite-variance-CLT case. I am curious if some way there is a situation that gCLT might not converge to ...
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### Large Set of Random Variables with Exponential Distribution

I'm struggling to understand how to solve the following problem. I have a random variable $X$ that represents the life time of a cellphone (in years) and I know that such variable follows an ...
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### Having a hard time with the law of the iterated logarithm

Let's say you have infinitely many i.i.d. Bernouilli variables $X_1, X_2, \cdots$ of parameter $p=\frac{1}{2}$. For instance, the binary digits of a random number. Let $S_n = X_1 + \cdots X_n$. The ...
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### How is central limit theorem applied?

All resources I find online state that when you sample from the population, the means form a normal distribution. I also found out that the "mean of the sample means equals the population mean". This ...
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### Why use sigma/sqrt(n) and not sigma^2/n?

My stats book says that according to CLT and if n is large, the distribution of means of random samples is approximately normal with mean = miu and variance = sigma^2/n, where sigma^2 is the variance ...
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### Asymptotic Distributions of form: $\sqrt{n}(\hat{\mu} - \mu, \hat{\sigma}^2 - \sigma^2)$

Suppose $X_1, \dots, X_n$ iid normals $N(\mu, \sigma^2)$, and $\hat{\mu}$ and $\hat{\sigma}^2$ are the MLE. How would one go about finding $$\sqrt{n}(\hat{\mu} - \mu, \hat{\sigma}^2 - \sigma^2).$$...
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### Rate of convergence for the sign of sample mean

What is the rate of convergence for $\textrm{sign}(\frac{1}{n}\sum_{i=1}^{n}X_{i})$? ( $X_{i}$ are independent and identically distributed as $F$ satisfying the conditions for central limit theorem)
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### Conditions to apply central limi theorem [duplicate]

Does the random variable need to have finite mean and variance in order to apply CLT?
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### Laplace approximation for small number of data

When a large number of data points is available, according to the central limit theorem, Laplace method can give an efficient, good approximation of posterior as a Gaussian distribution centered at ...
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### Calculating the confidence interval that with proability $x$, the population mean lies within $\pm D$ of the sample mean $m$, when $N > 100$

Suppose one has a series of $N$ ($N > 100$) data points sampled from a population with unknown distribution. $\mu$, the population mean, and $\sigma^2$, the population variance, are thus both ...
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### Central Limit Theorem & Sample Mean

This is a question which was given to me in an exam, and I'm confused about the approach taken in the Answer Key. The question is summarised as follows: A bank has 500 customers The total annual loan ...
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### Can the Central Limit Theorem be used in this case? [closed]

A student investigating study habits asks a simple random sample of 16 students at her large high school how many minutes they spent on their Math homework the previous night. Suppose the actual ...
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### Limiting distribution of sample variance and standard deviation

I have a centered Gaussian sample of $n$ elements $X_i,\,i=1,..,n$, with variance $\sigma^2$. I would like to find the limiting distribution of the sample variance \$\sigma_n^2=\frac 1n \sum_{i=1}^n ...