Random variables

In this section probability theory and random variables are introduced. Key results that underpin further sections are given here. If more information is required, the book Probability and Random Processes by Grimmett & Stirzaker (1982) is a good place to start.

Suppose a needle is dropped onto a floor made up of planks of wood. The needle may or may not intersect one of the joints between the planks. A single throw of the needle is called an experiment or trial. There are two possible outcomes. Either the needle intersects a joint(s) or it lands between the joints. By repeating the experiment a large number of times, the probability \(\mathbf{P}\) of a particular outcome or event can be calculated. Let \(A\) be the event "needle intersects a joint". Let \(N(A)\) be the number of occurrences of \(A\) over \(n\) trials. As \(n\to\infty\), \(N(A)/n\) converges to the probability that \(A\) occurs, \(\mathbf{P}(A)\), on any particular trial. On occasions, such as Buffon's needle throwing experiment (1777), a probability can be assigned to an outcome without experiment. The set of all possible outcomes of an experiment is called the sample space and is denoted by \(\Omega\). A random variable is a function \(X : \Omega\to\real\).

Uppercase letters will be used to represent generic random variables, whilst lowercase letters will be used to represent possible numerical values of these variables. To describe the probability of possible values of \(X\), consider the following definition. The distribution function of a random variable \(X\) is the function \(F_{X} : \real \to [0, 1]\) given by \(F_{X}(x) = \mathbf{P}(X \le x)\).


BUFFON, G. L. L. Comte de. Essai d'Arithmétique Morale. In: Supplément à l'Histoire Naturelle, v. 4. Paris: Imprimerie Royale (1777).

GRIMMETT, G. and STIRZAKER, D. Probability and Random Processes, Clarendon Press, Oxford (1982).