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For questions about the law of large numbers, a classical limit theorem in probability about the asymptotic behavior (in almost sure or in probability) of the average of random variables. To be used with the tags (tag:probability-theory) and (tag: limit-theorems).

The law of large number (lln) describes what happens if we perform an experiment a large number of times. It states that the average of the results obtained from a lot of trials will be close to the expected value. It also states that it will get closer to the expected value as more trials take place. The law guarantees a stable long-term results for the averages of events.

The strong law of large numbers states that the averages converge a.s., to the expected value \begin{equation*} \overline{X}_n\to \mu,~ n\to\infty . \end{equation*}

The weak law of large numbers states that the average converges in probability towards the expected value: \begin{equation*} \lim_{n\to\infty}Pr(|\overline{X}_n-\mu|>\epsilon)=0. \end{equation*}

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