The law of large numbers

The law of large numbers

The law of large numbers states that for a set of independent and identically distributed random variables, $X_1$, $X_2$, $\dots$ the following holds: \begin{equation} \lim_{n\rightarrow \infty} P\left( \left| \frac{S_n}{n} - \mathbb{E}(X) \right| > \epsilon \right) = 0 \nonumber \end{equation} where $n$ is the number of random variables, $S_n = X_1 + X_2 + \dots$ and $\epsilon$ is a small number. This expression cannot be used if your random variable has $\mathbb{E}(X) = \infty$

Syllabus Aims

  • You should be able to write down the law of large numbers without the proof.
  • You should be able to explain why the the expectation of random variables are useful by making reference to the law of large numbers.
  • You should be able to explain the random variables for which the law of large numbers does not hold.

Contact Details

School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN

Email: g.tribello@qub.ac.uk
Website: mywebsite