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 $H$ .