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).

819 questions
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How to apply the central Limit theorem?

I am interested on upper bounding the following probability as $n$ goes to infinity. \begin{equation} \mathbb{P} \left\lbrace \Big|( \xi_{n}- \mathbb{E} \xi_{n})\Big|> \ell \right\rbrace \end{...
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When to use the continuity correction for normal approximations of binomial probabilities.

so I'm confused as to when you actually use continuity correction. If a problem deals with a binomial distribution and we are asked to find probabilities using normal approximation (provided np>5 and ...
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LLN and CLT on $Y_n(t) := \boldsymbol{1}_{X_n \le t}$

Let $X_n$ be a sequence of i.i.d. random variables where $X_1$ has distribution function $F$. Fix $t \in \Bbb R$. Define $Y_n(t) := \boldsymbol{1}_{X_n \le t}$. Certainly, the $Y_n(t)$ are again i.i.d....
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Lindeberg's Condition is not necessary

Let $X_n^k$ be Triangular Array ($k\leq n$) of independent mean $0$ random variables. Suppose $\sum_k^n \text{Var}(X_n^k) = 1$. Lindeberg's Condition \begin{equation} \lim_{n\to\infty} \sum_k^n ...
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A failure of convergence of conditional distributions

Consider $n$ iid samples $(X_i,Y_i)$ generated such that $X_i$ is a truncated normal of mean $\mu_X$ truncated to the left of the origin, and $Y_i$ is a truncated normal of mean $\mu_Y$ truncated to ...
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Question about central limit theorem on two exercises.

I was presented two formulas, if random variables ${X_1,...,X_n}$ form a random sample of a distribution of mean $\mu$ and standard deviation $\sigma$ and $n \to \infty$: \begin{align} &P\left(\...
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falling blocks and the central limit theorem

I'm reading these notes and would like some help deriving the result at the bottom of page 2. Suppose that you are building a tower out of unit blocks. Blocks are falling from the sky, [increasing ...
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Is it possible to answer this problem with standard Central Limit Theorem or should we use Lindeberg-Feller CLT?

I have the following problem on my Statistics I problem set: Suppose that $X_t = \mu + U_t$, where $U_t = V_t + \rho V_{t-1}$ and $V_t$ are iid standard normal variables. Apply a CLT to ...
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Show that $\sqrt{\sum_{i}^n X_i} - \sqrt{n} \to Y \sim \mathcal{N}(\mu, \sigma^2)$ in distribution.

Let $(X_n)_n$ be a sequence of i.i.d. random variables with $\mathbb{E}X_1 = 1, \operatorname{Var}(X_1) = 3$. Show that $\sqrt{\sum_{i}^n X_i} - \sqrt{n} \to Y \sim \mathcal{N}(\mu, \sigma^2)$...
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Does the probability that at least one statistic will exhibit a “statistically significant” fluctuation converge to 1 as $N\rightarrow\infty$?

In the study and usage of statistics the idea that particular statistics will converge "almost certainly" to some value as the sample size $N$ diverges plays a key role (e.g. the central limit theorem,...
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How does the first moment affect convergence in distribution?

I have the following problem on my Statistics Problem Set: I was able to use the multivariate Delta Method and solve items a) and b). However, I am struggling with items c) and d). Let me present my ...
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Interpreting estimated percentage as a distribution

Imagine I am trying to determine the percentage $p$ of people in the US who voted for the democrats (or republicans, if you prefer). I can determine this by the following process: Randomly ...
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Multidimensional CLT for Exchangeable Random Variables

From Wikipedia Let $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ be independent $d$ dimensional random vectors having $0$ mean. Let $\mathbf{S}=\sum_{i=1}^n \mathbf{X}_i$ and $\mathbf{\Sigma}$ be ...
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Help finding central limit theorem approximations - Normal Distribution Equation

I was given f(x)=|x| as a probability distribution. I've summed the results of a Monte Carlo with N terms and plotted a thousand of these sums in a normalized histogram. Now I need to compare this ...
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Central limit theorem for dependent random variables with covariance condition

Consider a sequence of identically distributed real-valued random variables $(X_i)_{i\in\mathbb{N}}$, with $\mathbb{E}\left[X_i\right]=0$ and $\mathbb{E}\left[X_i^2\right]=1$. Suppose that there ...
This is another statistical question that I cannot fully understand: Suppose that $100$ fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Now that the ...
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 ...