# 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). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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### Sum of independent but not identically distributed uniform random variables

Let $(X_{j})_{j\geq1}$ be independent and uniformly distributed on $(-j,j)$ and let $S_{n} = X_{1} + ... + X_{n}$. Show that $\lim_{n \to \infty}S_{n}/n^{3/2}=Z$ in distribution where $Z$ ~ $N(0,1/9)$ ...
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### When can the level of the test be exactly $\alpha ?$ in non randomized test. And how to use CLT to find the critical value.

Let $X_{1}, \ldots, X_{n}$ be a sample from the Bernoulli distribution with parameter $p$ Consider testing $H_{0}: p=p_{0}$ versus $H_{1}: p=p_{1}$ where $p_{0}<p_{1}$ are known numbers. (a) Using ...
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### How to determine confidence interval using central limit theorem?

How to determine the confidence interval for the unknown theta parameter of the Uniform($[0, \theta]$)-distribution using the central limit theorem, considering that the significance level is given ...
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### Convergence in probability from the central limit theorem

My question comes from the proof of the Delta method. One of the conditions states that $\sqrt{n}(Y_n - \theta) \rightarrow N(0,\sigma^2)$ in distribution for some sequence of random variables $Y_n$. ...
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### What is a specific observation of a probability parameter estimate? Programmer trying to understand statistics

I am trying to understand the following statement about a collection of independent and identically distributed Bernoulli random variables. We have $\theta$ as the probability of success of the ...
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How can I find the probability $$\lim_{n \to \infty} P\left(\sum_{i=1}^n X_i>0\right) = \lim_{n \to \infty} P\left(\frac{1}{n}\sum_{i=1}^n X_i>0\right)$$ if $X_i$ are iid? If the expectation of $... 0answers 20 views ### CLT for weighted sum of Bernoulli Variables Suppose I have the random variable$y_i = w_i x_i z_i$, with$z_i$beeing a Bernoulli variable,$x_i$is drawn from a unimodal distribution with finite variance, and$w_i$are constants in some ... 1answer 50 views ### Sums of binomial coefficients - accuracy of CLT approximation? I was trying to figure out the asymptotics of the sum $$S(n) = \sum_{k=0}^{\lfloor \alpha n \rfloor} \binom{n}{k}$$ for$0 < \alpha < 1/2$(actually, the specific case of interest for me is$\...
I am trying to follow the CLT proof of Wikipedia's article on CLT I can follow up to and including the part that $Z_{n} \to N(0,1)$ as $n \to \infty$. I cannot understand the very last sentence of ...