Law of Large Numbers

For a sequence of iid random variables \(X_1,\ldots,X_n\) let the sample mean be defined as,

\[ \bar{X} = \frac{1}{n}\sum{X_i} \]

As \(n \rightarrow \infty\), the probability that the sample mean deviates from the population mean \(\mu\) absolutely by some positive error \(\epsilon\) tends to,

\[ P\left[\abs{\bar{X}- \mu} < \epsilon \right] \rightarrow 1 \]