The Law of Large Numbers

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 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

$$ \frac{1} {n}\sum\limits_{k = 1}^n {X_k \to p{\text{ }}as{\text{ }}n \to \infty ,} $$

where p = P(X₁ = 1) is the success probability.