# Questions tagged [central-limit-theorem]

"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](http://en.wikipedia.org/wiki/Central_limit_theorem)).

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### What will be the confidence interval(for below question) for 99 percent confidence level [68 95 99 rule] [on hold]

Is the below solution correct....because the probability of obtaining a sample which would have the mean height of 70 will fall under the intervals calculated using the mean+/- 3(SD) [99% Confidence ...
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### Central limit theorem for resampling

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic? As a simple example, suppose there are $n$ i....
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### Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
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### central limit theorem: do we care about standard deviation within one sample of size n? [duplicate]

I'm learning applications on Central Limit Theorem and got really confused with a few points. Think of an example of applying Central Limit Theorem: We have a whole population of 10 billion items ...
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### Do we use the SD of whole population or SD of just one sample to calculate SE of samples means in central limit theorem?

I'm learning applications on Central Limit Theorem and got really confused with a few points. According to this tutorial, the procedure to apply CLT usually goes like this: So if ...
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### Self-study: Which option is NOT a prediction of the central limit theorem?

"All of the following are predictions of the Central Limit Theorem except: 1) The sample mean distribution will be approximately normally distributed if the sample size is large 2) The mean of the ...
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### Understanding regression to mean

As far as I can understand regression to mean is that if I make a measurement, say the mean of the test scores of some students, when the measurement is repeated with the same student with same ...
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### How to find the “variance” when using “central limit theorem” on a Poisson distribution? [closed]

Assume we have N number of inventors in a company. Inventor i expects to invent X_i number of inventions per year. How many inventions each of them invent per year has a "Poisson distribution" where ...
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### Using CLT for hypothesis testing

I have two not normal distributions (~1k samples in each), looks like exponential: So, I need to check its means, that's why I have following questions: The easiest way to do it - is to use Mann-...
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### Why does increasing the sample size of coin flips not improve the normal curve approximation?

I'm reading the Statistics (Freeman, Pisani, Purves) book and I'm trying to reproduce an example where a coin is tossed say 50 times, the number of heads counted and this is repeated say 1,000 times. ...
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### Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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### How to prove $\mathbb{E}[\left | \frac{1}{n} \sum X_i-\mu \right |]=\mathcal{O}(\frac{1}{\sqrt{n}})$ as $n \rightarrow \infty$?

$X_i$ is iid sequence with mean $\mu$ and finite variance $\sigma^2$.
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### Distribution for average of multiple binomial proportions

Assume we have a population $N$ and a proportion $p$ of that population with a characteristic of interest. Both $N$ and $p$ are unknown. Furthermore, assume that we have $k$ random samples $(n_i, x_i)$...
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### How to calculate the mean of a sub sample, given the mean of the super sample and the standard deviation of the population?

I need to run a simulation of cash flows for a project. We are selling a service. The service comes with a range of options. Depending on the specific options chosen, the cost of the service can ...
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### Central limit theorem (CLT) writing

Is there a reason why we are used to write the CLT as $\sqrt{n}(\overline{X}_n-\mu)\stackrel{d}{\rightarrow}N(0,\sigma^2)$ and not as $\overline{X}_n\stackrel{d}{\rightarrow}N(\mu, \frac{\sigma^2}{n})$...
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### Parametric Bootstrap Central limit theorem non i i d

I am having paired data with missing values in a single arm. I am willing to use parametric bootstrap with specific quadratic tests to test the hypothesis of equality of means. My model is as follows:...
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### Question about CLT proof

I'm working through the CLT proof on Wikipedia trying to get a better intuition, and it made me wonder what an individual distribution looks like after dropping the o(t^2/n) terms from the Taylor ...
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### Can the Berry-Esseen theorem tell us whether acceptable inference may be achieved by parametric tests?

I refer in particular to such choices: 1) t-test (or its generalizations: ANOVA or Hotelling's $t^2$) vs its non-parametric alternatives (e.g., U Mann-Whitney test and its generalizations); 2) Pearson'...
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### Trouble relating the Central Limit Theorem to confidence intervals

I'm having trouble understanding how the Central Limit Theorem (CLT) implies that we can create confidence intervals as we do. For example, Slide 5 from these lecture notes essentially lays out the ...