Probability: A Graduate Course pp 265-327 | Cite as

# The Law of Large Numbers

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## Abstract

We have mentioned (more than once) that the basis for probabilistic modeling is the stabilization of the relative frequencies. Mathematically this phenomenon can be formulated as follows: Suppose that we perform independent repetitions of an experiment, and let where

*X*_{k}= 1 if round*k*is successful and 0 otherwise,*k*≥ 1. The relative frequency of successes is described by the arithmetic mean, \( \frac{1} {n}\sum\limits_{k = 1}^n {X_k \to p{\text{ }}as{\text{ }}n \to \infty ,} \) and the stabilization of the relative frequencies corresponds to$$
\frac{1}
{n}\sum\limits_{k = 1}^n {X_k \to p{\text{ }}as{\text{ }}n \to \infty ,}
$$

*p*=*P*(*X*_{1}= 1) is the success probability.## Keywords

Independent Random Variable Complete Convergence Uniform Integrability Partial Maximum Common Density
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, Inc. 2005