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.
Description and link | Module | Author | ||
| Some notes on the law of large numbers and the central limit theorem. These notes include proofs for these two theorems. | SOR3012 | J. F. McCann |
Description and link | Module | Author | ||
| Introducing the law of large numbers | SOR3012 | G. Tribello | ||
| A video explaining how the law of large numbers is used when calculating a histogram | SOR3012 | G. Tribello |
Contact Details
School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN
Email: g.tribello@qub.ac.uk
Website: mywebsite