# 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).

791 questions
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### What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [on hold]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
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### Emergence of Lognormal distribution for the concentration of chemical compounds

I'm currently reviewing the literature about lognormal distributions describing/approximating the variability of a given chemical compound across different cells/ samples etc etc. The main argument ...
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### Is Confidence Interval taken on one Random Sample or A Sampling Distribution

I am stuck at CI. Intuitively, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data ( random samples), that might contain the true value (mean) ...
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### Doubts regarding Central Limit Theorem

All, I am studying CLT and I had a doubt. I understand that sampling distribution has a mean same as population mean and std.dev as 6/sqrt(n) where n is sample size. Each point on X axis of ...
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### How to compute $\lim_{n \to \infty}P(C_n>C_0)$?

The unit price of a certain commodity evolves randomly from day to day with a general downward drift but with an occasional upward jump when some unforeseen event excites the markets. Long term ...
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### How to handle little $o$ in the central limit theorem

I am having some trouble understanding a couple of lines in the proof of the central limit theorem using characteristic functions: https://en.wikipedia.org/wiki/Central_limit_theorem#...
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### Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof. Here is what we are given: ...
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### Proving IID Central Limit Theorem using Lindeberg Conditions.

The goal is to prove the IID Central Limit Theorem through Lindeberg's Condition. Suppose that $X_1,X_2,\ldots\displaystyle\sim\text{i.i.d.}$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^2<\infty$. ...
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### A die is thrown until the first time the total sum of the face values of the die is 700 or greater. What is the probability for $n$ tosses?

Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required We're supposed to be using ...
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### Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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### Show $Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$

Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly ...
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### If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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### Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
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### An example where Lyapounov Condition is much easier than the Lindeberg?

Currently studying the Lindeberg-Levy-Feller and Lyapounov's Central Limit Theorems, and was wondering - for sake of justification - are there any examples where verifying Lyapounov's condition is ...
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### A central limit theorem for dependent random variable.

Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e. $$u_j\stackrel{d}{=}\text{N}(0,1).$$ Call $\mu_r=\mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution ...
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### CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...