# Questions tagged [central-limit-theorem]

This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems).

820 questions
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### Calculation of Standard Deviation of Sample Mean

This is another statistics problem that I have, which I cannot make sense of: An IQ-test is normal to $\mu = 100$ and $\sigma = 10$. What is the standard deviation of the sample mean of a sample ...
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### bayesian hyperpriors central limit theorem analogy

Let's say I have some distribution F, and then put a prior on a parameter of that distribution by using some prior distribution P1. Then let's say I put a hyperprior on that hyperparamter by using a ...
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### Convergence in Distribution of Sums of Random Variable

Suppose I have $X_1,X_2,...,X_n$ random variables that are independent and identically distributed, from ANY distribution. Suppose that $E(X_i)=\mu$ and $V(X_i)=\sigma^2$. Suppose I define the ...
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### Problem when Multiplying Sample Distributions

Question: "An aeroplane is licensed to carry 100 passengers. If the weights of passengers are normally distributed with a mean of 80 kg and a standard deviation of 20 kg.Find the probability ...
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### Proving Multivariate Central Limit Theorem using Lindeberg Theorem

I'm reading a proof of Multivariate CLT using Lindeberg Theorem. Let $X_n = (X_{ni},... ,X_{nk})$ be independent random vectors all having the same distribution. Suppose that $E[X_{nu}]<\infty$; ...
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### Convergence of Bernoulli RV to Normal Distribution

Suppose we have $X_1,X_2,...$ Bernoulli RV (not necessarily independent) with $P(X_i=1)=p_i$. I am trying to understand what conditions are sufficient in order for a CLT to hold. Surely if the ...
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### Central Limit Theorem of an expectancy

Let $S_n(\psi)$ be a uniformly bounded statistic that depends on a real-valued random variable $\psi$, whose the distribution $F_{\psi}$ is known. I established using the Central Limit Theorem that ...
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### Convergence in distribution, CLT (solution check)

$(X_n), (Y_n)$ independent sequences of iid random variables such that $EX_1=EY_1=0$, $\operatorname{Var}(X_1)=\operatorname{Var}(X_2)=\sigma^2$. $S_n=\sum_{k=1}^nX_k$, $T_n=\sum_{k=1}^nY_k$. 1) ...
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### Determine limit distribution

Let $X_1,\dots,X_n$ be an iid sample from a uniform distribution $U[-\theta,\theta]$ for some parameter $\theta>0$. Define $X_{(n)}=\max\{X_i: 1\leq i\leq n\}$ and to estimate $\theta$ consider the ...
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### Central Limit Theorem and Sum of Random Variables

Let each $X_i$ be i.i.d and $Y_i=\ln(X_i)$ is exponentially distributed with mean $\alpha^{-1}$ and variance $\alpha^{-2}$. Now let $$S_n=\left[ \prod_{i=1}^{n}X_i \right ]^{n^{-1}}.$$ Use the ...
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### Help verifying the Lindeberg condition

Suppose $X_1, X_2,...$ is a sequence of independent, identically distributed random variables with $\mathbb{E}[X_n] = 0$ and $\mathbb{E}[X_n^2] = 1$. Find sequences of constants $a_n$ and $b_n$ ...
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### Central limit theorem with different variance

Let $\{ X_n, n\geq 1 \}$ be a sequence of independent random variables with $P(X_n=n^{\lambda})=P(X_n=-n^{\lambda})=\frac{1}{2}$ where $0<\lambda<\infty$ Show that the central limit theorem is ...
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