In probability theory, the *law or large numbers* states that the empirical mean of a process where the random variables are independent and identically distributed tends to its expectation value.

It can be seen as a mathematical formalization of the idea that, even in presence of randomness, the *average behavior* of a system tends to be predictable, and the larger the sample size is, the better the prediction is.

For example, when rolling a fair die repeatedly, the average of the number rolled tends to

$\frac{1+2+3+4+5+6}{6} \;=\; 3.5 .$

The term *law of large numbers* refers to a few different statements, depending on the particular convergence of random variables? considered.

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- probability theory, categorical probability
- random variable, expectation value
- empirical mean, empirical distribution
- iid random variables, de Finetti's theorem, zero-one law
- central limit theorem, Glivenko-Cantelli theorem?
- ergodicity, ergodic theorem?
- martingale convergence theorem?
- asymptotic equipartition property?

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category: probability

Last revised on July 21, 2024 at 14:16:49. See the history of this page for a list of all contributions to it.