1.1 Definitions

Definition 1.1 (Sample space) A sample space, often denoted by \(S\), is the set of all possible outcomes of a random trial or experiment.
Definition 1.2 (Random) An event is random if its outcome is uncertain, and is (at least partially) due to chance.
Definition 1.3 (Variable) A variable changes its value from situation to situation, or from realisation to realisation. That is, the value varies.

Example 1.1 (Variables; sample space) The country in which people were born is a variable; it varies from person to person.

The sample space is rather large, but we could use this more restricted sample space: \[ S = \{ \text{Australia}, \text{UK}, \text{China}, \text{Canada}, \text{Elsewhere}\}. \]

Definition 1.4 (Random variable) A random variable (often written as ) is a ‘rule’ associating a number with each outcome in a sample space \(S\).

Mathematically: a rv is a function whose domain is the sample space \(S\), and whose range is a set of real numbers.
Example 1.2 (Random variables) We could (arbitrarily) define a rv, say \(C\), such that \[ C = \left\{ \begin{array}{lr} 1 & \text{for Australia}\\ 2 & \text{for UK}\\ 3 & \text{for China}\\ 4 & \text{for Canada}\\ 5 & \text{for elsewhere} \end{array} \right. \]

A rv is often denoted by a capital letter (for example, \(X\)), and lower case letters used (for example, \(x\)) for the value that the rv takes.

So \(\Pr(X=x)\) means “the probability that the rv \(X\) takes a particular value \(x\).”

Sometimes, we just write \(x\) for both when there is no ambiguity.

Example 1.3 (Random variables) Suppose we are examing delicate light bulbs, and keep trying them until we find one that fails.

The rv \(X\) can be defined as ‘the number of bulbs we try until we find one that fails.’

The sample space is \(S = \{1, 2, 3, \dots\}\). Notice that there’s no upper limit in theory.

If we write \(\Pr(X > 4)\), this means ‘the probability that more than four trials are needed to find a bulb that fails.’
Example 1.4 (Random variables) The heights of adult Australian females in centimetres is a random variable, say \(H\).