# Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "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)

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### Non-normal sample from a non-normal population (option returns) does the central limit theorem hold?

I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the ...
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### CLT so we have enough burgers!

An event has 32 people. Caterers need to make burgers for these people. They expect that a person at this event might need 0, 1 or 2 burgers with probabilities 0.2, 0.55, and 0.25 respectively. We ...
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### Picking a specific estimated CDF from a set of CDFs provided by an ECDF

Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...
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### Convergence of variance of sample median, pt. 2

Follow on question to this, answered negatively by Thomas Lumley. We reprint it here for convenience. In this SE question, it is stated that there is a central limit theorem for the sample median, ...
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### Cental limit theorem, Chebyshev's inequality, and convergence of distributions through rescaling

I've been thinking about this issue for a few days and although read some of relevant questions on this site, still couldn't get it off my mind. Suppose we have $n$ i.i.d random variables $X_i$ with ...
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### Limit of $\mathbb{P}(\sqrt n - 1 \leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt{n} + 1)$ wrt $n$ for standard multivariate normal $X$

Suppose $X$ is a standard multivariate normal distribution. Then what is the limit of the $c_n$ where the $c_n$ are $\mathbb{P}(\sqrt n - 1\leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt n + 1)$? I want ...
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### Sample and population: definition for workplace statistics?

This might be a bit of a dunce's question, but I was wondering about the difference between a sample and a population. Obviously, if you have data relating to 200,000 people but you only look at 200 ...
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### Convergence of variance of sample median

In this SE question, it is stated that there is a central limit theorem for the sample median, namely $$\sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}),$$ as $n\to\infty$ where $Y_n$ is the ...
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### Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
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### Does a Binomial converge to Poisson or Normal?

I have read the answer here. Here the distinction is that If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson ...
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### Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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### Is bootstrapping redundant for computing confidence intervals for the sample mean in large samples?

If we are just interested in computing confidence intervals for the population mean $\mu$ using a sample $X_1,X_2,\dots,X_n$ of $n$ iid random variables is bootstrapping redundant if $n$ is large? I ...
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### CLT for non iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$. I'm looking for CLT of the sum $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking ...
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### Differences Between the Central Limit Theorem and Consistency

I have recently finished studying the central limit theorem and the idea of consistency. I am still a little fuzzy about them, so I was wondering what are some key similarities and differences of the ...
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### Approximating $E[g(\overline X_n)]$ and want to bound the remainder using some form of CLT or Berry-Essen Theorem

If we have a set $X_1,\dots,X_n$ of iid random variables with finite mean $\mu$ and variance $\sigma$, the CLT says that $\sqrt{n}(\overline X_n - \mu) \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$. If ...
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### what is the standard deviation of the geometric mean sample distribution?

I wrote a python script to take a population distribution of a random variable in the interval (0,1) to be uniform and make 2 sample distribution: The fist is the distribution of the arithmetic mean ...
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### confidence interval of $\beta$, where $X$'s are from exponential distribution

Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$. It can ...
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### How to think about confidence intervals in the context of linear regression?

I think I understand confidence intervals for sampling distributions, but am trying to connect this understanding to the confidence bands I see around linear regression lines. According to the ...
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### Probability of averages

I have a random variable $Y$ and I am taking an independent sample of $n$ from this RV. I'll refer to this sample as $Y_n$, and I define the average of this sample as $\bar{Y}_n$. The maximum of this ...
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### Which distributions originates from sampling near boundaries [closed]

I have a certain non-deterministic process which receives a parameter k that belongs to a fixed domain [a, b]. I am able to generate samples from the output of the process in the domain of k and by ...
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### Central Limit Theorem Approximation and Relation to Law of Large Numbers

Assume the Linberg-Levy CLT to where we know $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{d}N(0,1).$$ I feel like I commonly see then that $$\bar{X}_n\approx N(\mu,\frac{\sigma^2}{n}),$$ but ...
Let $D$ be a distribution with finite mean $\mu$ and finite variance $\sigma^2$. Consider the distribution $S_n$ of the sample mean of $n$ i.i.d. values from $D$. I understand that the Central Limit ...