# Questions tagged [law-of-large-numbers]

Several theorems stating that sample mean converges to the expected value as $n\to\infty$. There is a weak law and a strong law of large numbers.

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### Law of large numbers and standard deviation

In wikipedia, the law of large numbers is defined as follows : "The average of the results obtained from a large number of trials should be close to the expected value and tends to become closer ...
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### How does the law of large numbers relate to regression to the mean?

Intuitively, these two important statistical principles appear to describe two facets of the same phenomenon, namely that in the long run, any extreme occurrences get counter-balanced, and things tend ...
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### Does coin tossing obey mean reversion? [duplicate]

If a coin comes up Heads 90 times out of the first 100 tosses, should one expect Tails to make a comeback over the next 100? Reference from this page: https://www.financialwisdomforum.org/gummy-stuff/...
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### Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely?

Edit Sep 19 this answer on Mathoverflow matches simulation results Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges ...
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### Sum of squared normals: $\sum_{k=1}^{n}U_k^2 = \frac{1}{n}\sum_{k=1}^{n}a_k^2 \cdot \Gamma\left(\frac{n}{2},\frac{1}{2}\right)$

Assume $U_k \sim \mathcal{N}(0,a_k^2)$, where $a_k \rightarrow c > 0$ as $k \rightarrow \infty$. It follows that $U_k^2 \sim \Gamma(\frac{1}{2}, \frac{1}{2a_k^2})$. I'm interested in the exact and ...
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### Does central limit theorem help in making inference only about the population mean and not other parameters?

As per the weak law of large numbers, if your sample size is large, your mean of the sample is likely to be closer to the population mean than in a smaller sample. Additionally the CLT tells us the ...
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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 ...
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### Weird behavior in simulating LLN for standard deviations in R

I am simulating the law of large numbers and how it applies to standard deviation, as well. I wrote this code that works well but there is something that I am having a hard time understanding. ...
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### Confused about conditions of the weak and strong laws of large numbers

I am a little confused by what conditions need to hold for the weak law of large numbers (WLLN) and the strong law of large numbers (SLLN) to be true. It seems different sources give me different ...
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### Terminology - population and sample in repeated rolling dice experiments

The the law of large numbers wikipedia page has an example on fair, six-sided dice: According to the law of large numbers, if a large number of fair six-sided dice are rolled, the average of their ...
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