Law of Large Numbers

Definition

Weak Law of Large Numbers

Consider a Sequence $\{X_n\}$ of i.i.d. random variables with mean $E[X_i]=\mu$, variance ${\sigma }^2<\infty$. Let ${\bar{X}}_n=\frac{1}{n}\sum _{i=1}^n{X}_i$ be the sample average. Then, the sample average converges in probability to the expected value. ${\lim }_{n\to \infty }{\bar{X}}_n\stackrel{P}{\to }\mu$

Strong Law of Large Numbers

Consider a Sequence $\{X_n\}$ of i.i.d. random variables with mean $E[X_i]=\mu$, variance ${\sigma }^2<\infty$. Let ${\bar{X}}_n=\frac{1}{n}\sum _{i=1}^n{X}_i$ be the sample average. Then, the sample average converges almost surely to the expected value. ${\lim }_{n\to \infty }{\bar{X}}_n\stackrel{a.s.}{\to }\mu$