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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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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22 views

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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36 views

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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31 views

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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How can I find the asymptotic relative efficiency of two quantities, estimating $\sigma$?

Let $X_1,...,X_n$ be a random sample from $N(0,\sigma^2)$, where $\sigma>0$ is unknown. We try to estimate $\sigma$ using $T_1=\sqrt{\frac{\pi}{2}}\frac{1}{n}\sum^n_{i=1}|X_i|$ and $T_2=\sqrt{\frac{...
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How to compare the results of Monte Carlo simulations with different sample sizes?

I'm simulating the Monopoly game and trying to find out the most frequent squares in the board. For further details, it is Project Euler problem 84. The solution will not be spoiled. To illustrate ...
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56 views

Showing Lyapunov (Lindeberg) condition holds for sum of independent bernoulli distribution with poisson tail probability

I'm trying to find an asymptotic distribution of the follwing random variable $$Z_n=\sum_{i=0}^n Y_i$$ where $Y_i = I[T_i<t]$ with $T_i \text{~} Gamma(i, \lambda)$. Here $t$ is a fixed number. ...
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What is the difference between a non-central limit theorem and the usual central limit theorems?

I'm reading a paper where the authors prove the following theorem. They then say that this constitutes a non-central limit theorem for the variables in question. Since I have never heard this term (...
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51 views

Delta method confusion

I am supposed to use the delta method to find the limiting distribution for $$\sqrt{n}\left(\frac{\overline{X}_n}{1-\overline{X}_n} - \frac{E(X)}{1-E(X)}\right)$$ where $f(x, \theta)=\theta x^{\theta-...
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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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What distribution does the mean of a random sample from a Uniform distribution follow?

For example, let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=1,\theta-1/2 < x < \theta +1/2$. Clearly, $X_i \sim U(\theta-1/2 , \theta +1/2)$. Some intuition would suggest that $\bar{X}...
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Limiting distribution of infinite sparse sum

Let $N$ be a positive integer. I consider $N$ random variables $X_1^{(N)}, X_2^{(N)}, \dots, X_N^{(N)}$, all independent and identically distributed, each taking values $\pm 1$ with probabilities $p/(...
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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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51 views

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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61 views

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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68 views

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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How are the variance of an estimate to $\int_Bf\:{\rm }d\mu$ and the deviation of $f$ from the mean $\frac1{\mu(B)}\int_Bf\:{\rm d}\mu$ related?

Let $(E,\mathcal E,\mu)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $\mu$ and $$A_nf:=\frac1n\...
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Confidence in Negative Binomial Prediction

I have predicted future demand of stock from observed historical demand projecting a negative binomial demand curve. Demand is extrapolated to multiple periods using this approach. I'm pretty happy ...
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How close a sample is to the Normal distribution ( Berry-Esseen Theorem)

My question is how can I use the Berry-Esseen Theorem to know how close to the Gaussian distribution is $L$, where $$L=nLn(2)+Ln(r_1)+Ln(r_2)...Ln(r_n).$$ $r_i \geq 0$ is a i.i.d. random variable ...
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105 views

Should bootstrapping and collecting sample means from a series of binomial distributions result in standard normal?

I’m trying to better understand several statistical concepts (bootstrapping, central limit theorem, and confidence intervals) by applying them to a binomial distribution (you can think of it as a coin ...
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Question about Central Limit Theorem with an example

“If height in centimeters in some given population is distributed with mean 175 and standard deviation 10, the central limit theorem implies that the probability of finding a person taller than 177 ...
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Asymptotics of 2 x 2 precision matrix

Edited to give the answer... but I still don't understand where it came from! Suppose we have $$X_1, X_2,..., X_n \overset{i.i.d.}{\sim} N(0, \Omega^{-1})$$ where $\Omega \in \mathbb{R}^{2 \times 2}$ ...
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Covariance of conditional poisson random variable sequence

Suppose $X_0,X_1,\cdots$ are iid $Poisson(\theta)$ r.v. Define $Y_k = X_k I_{\{ X_{k-1} = 0 \}}$ for $k=1,2,3,\cdots$ Find the limit of $Var(\sqrt{n}\overline{Y_n})$ and asymptotic distribution of $...
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Independent samples test for mean without normal distribution with T-score

I have two independent samples of male and female order value. Female number of observations = 26887 Male number of observations = 12928 Female mean order value = 133.03 Male mean order value = 145....
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Estimation and hypothesis testing for the difference in squared bias for two random variables

My Question: Let $X_t$ and $Y_t$ denote two time-series random variables, both of which are estimates of the random variable $\theta_t$. Let $U_t = X_t - \theta_t$, and $V_t = Y_t - \theta_t$. The ...
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Normal distribution in nature: additive result of multiple variables? [duplicate]

We found normal distribution is so common in nature, such as many measurement of species (weight, height or size). From the point of central Limit Theorem, Can I intuitively understand this as the ...
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24 views

Standardisation and Central Limit Theorem

i got a question about standardization: I have to calculate $P(\bar{X}<2,2)$ with $n = 100$ ; $E[X_i] = E[\bar{X}] = 2$ and $V[X_i] = 4$ $(-> V[\bar{X}] = 0.04)$ I used the CLT: $P(\frac{\bar{...
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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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1answer
29 views

Clarification of Notation

I am learning some new statistics concepts, and I am having a bit of difficulty discerning certain notation. How does $\hat{\sigma}$ differ from $\sigma$ in the description below: CLT: If $X_1,...,...
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How is the G-test related to the Central Limit Theorem?

I remember having read that the G-test is a more accurate version of Pearson's $\chi^2$-test, and that both are derived directly from the Central Limit Theorem. Unfortunately I cannot remember where ...
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Binomial distibution with mean 0

I give the following context. Test hypothesis $H_0:p=p_o$ for Bernoulli distributed random variable $X$, with parameter $p$ and samples $X_i$. For estimate $\hat{p}=\frac{1}{n}\sum_{k=1}^n X_i$, $n\...
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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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58 views

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 ...
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Confidence intervals for population

These might be slightly basic questions for confidence intervals but I can't think exactly how to resolve them. Considering an example where I have access to the entire population e.g. the annual ...
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What is the limiting posterior in the generalized Bayesian central limit theorem?

The central limit theorem characterizes the limiting distribution of the sum of increasingly many finite-variance independent random variables: the limit is Gaussian. The generalized central limit ...

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