2.1 Random variables

A random variable is an indicator of the outcome of a probability experiment that always outputs numerical values. i.e. it is a function from the sample space, $\mathrm{\Omega }$, to the real line, $ℝ$. The theory is richer than that of ordinary events because of the additional structure imposed by the number system.

Consider the previous example of a coin tossed twice with $\mathrm{\Omega }=\{\{HH\},\{HT\},\{TH\},\{TT\}\}$. We might define $R=0$ when the outcome is $TT$, $R=1$ when the outcome is $HT$ or $TH$, and $R=2$ when the outcome is $HH$.

NB: A random variable is NOT a number, but is something that outputs values that are numbers, i.e. a function. In the above example $R(\{TT\})=0$, for example.