# Tag Info

Accepted

### Are there any examples of where the central limit theorem does not hold?

To understand this, you need to first state a version of the Central Limit Theorem. Here's the "typical" statement of the central limit theorem: Lindeberg–Lévy CLT. Suppose ${X_1, X_2, \dots}$ is ...
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### Expectation of 500 coin flips after 500 realizations

If you "know" that the coin is fair then we still expect the long run proportion of heads to tend to $0.5$. This is not to say that we should expect more (than 50%) of the next flips to be ...
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### Debunking wrong CLT statement

As whuber notes, you can always point your collaborators to a binary discrete distribution. But they might consider that "cheating" and retreat to the weaker claim that the proposed ...
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### Debunking wrong CLT statement

This is quite a ubiquitous misunderstanding of the central limit theorem, which I have also encountered in my statistical teaching. Over the years I have encountered this problem so often that I have ...
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### Why does increasing the sample size of coin flips not improve the normal curve approximation?

In the second case, by increasing the number of tosses, you increase the number of bins a single trial can fall into. While the first case of experiment 2 only has a maximum of 100 bins that can be ...
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### How can the central limit theorem hold for distributions which have limits on the random variable?

This is an excellent question, since it shows that you are thinking about the intuitive aspects of the theorems you are learning. That puts you ahead of most students who learn the CLT. Here I will ...
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### Does this code demonstrate the central limit theorem?

Here's a complete study in a few lines. For a given set of sample sizes n and underlying distribution r, it generates ...
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### Are there any examples of where the central limit theorem does not hold?

Although I'm pretty sure that it has been answered before, here's another one: There are several versions of the central limit theorem, the most general being that given arbitrary probability density ...
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### Why does Central Limit Theorem break down in my simulation?

Let's recall, precisely, what the central limit theorem says. If $X_1, X_2, \cdots, X_k$ are independent and identically distributed random variables with (shared) mean $\mu$ and standard deviation ...
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### Debunking wrong CLT statement

The central limit theorem states that the mean of the data will become normally distributed as the sample size increases, it says nothing about the data itself. Another way to put it is the ...
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### How is CLT related to the condition of data (normality assumption)?

CLT states that if sample is big enough, our data approximates normal distribution That is false. If you take a very large sample from a non-normally distributed population, the empirical ...
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### Why is the limit of a Chi squared distribution a normal distribution?

This property follows from the central limit theorem, using the fact that the chi-squared distribution is obtained as the distribution of a sum of squares of independent standard normal random ...
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### Expectation of 500 coin flips after 500 realizations

The law of large numbers doesn't state that some force will bring the results back to the mean. It states that as the number of trials increases the fluctuations will become less and less significant....
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### Are there any examples of where the central limit theorem does not hold?

No, CLT always holds when its assumptions hold. Qualifications such as "in most situations" are informal references to the conditions under which CLT should be applied. For instance, a linear ...
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### Why a sample of skewed normal distribution is not normal?

I was under the impression that if I randomly sample from a skewed normal distribution, the distribution of my sample would be normal based on central limit theorem You are incorrect in your ...
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### How can we get a normal distribution as $n \to \infty$ if the range of values of our random variable is bounded?

Here is what you are missing. The asymptotic distribution is not of $\bar{X}_n$ (the sample mean), but of $\sqrt{n}(\bar{X}_n - \theta)$, where $\theta$ is the mean of $X$. Let $X_1, X_2, \dots$ be ...
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### What intuitive explanation is there for the central limit theorem?

This answer hopes to give an intuitive meaning of the central limit theorem, using simple calculus techniques (Taylor expansion of order 3). Here is the outline: What the CLT says An intuitive proof ...
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### Normally distributed errors and the central limit theorem

This may be better appreciated by expressing the result of CLT in terms of sums of iid random variables. We have $$\sqrt{n} \frac{ \bar{X} -\mu}{\sigma} \sim N(0, 1) \quad \text{asymptotically}$$ ...
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### Expectation of square root of sum of independent squared uniform random variables

One approach is to first calculate the moment generating function (mgf) of $Y_n$ defined by $Y_n = U_1^2 + \dotsm + U_n^2$ where the $U_i, i=1,\dotsc, n$ is independent and identically distributed ...
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### Why does Central Limit Theorem break down in my simulation?

In general, the size of each sample should be more than $5$ for the CLT approximation to be good. A rule of thumb is a sample of size $30$ or more. But, with the population of your first example, $5$ ...
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