Law of large numbers

For a sequence of independent, identically distributed (iid) random variables with finite mean $E(X)=\mu$ and sample mean $M_n=\frac{1}{n}\sum X_i$, the weak law of large numbers states:

$$\underset{n\to \infty }{\lim }P(|M_n-\mu |<ϵ)=1$$

i.e., for a large enough sequence of $n$, the sample mean will be close to the true mean with high probability.

The strong law of large numbers states that for a sequence of iid RVs with finite mean and finite variance:

$$P\left(\underset{n\to \infty }{\lim }{M}_n=\mu \right)=1$$

i.e., the sequence of sample mean calculations will eventually converge to $E(X)=\mu$. Both the strong and weak laws are roughly equivalent.

For computations, we should calculate with:

$$P(|M_n-\mu |<ϵ)\ge 1-\frac{{\sigma }^2}{n{ϵ}^2}$$