Function f of the set of possible outcomes of a random experiment in a subset of the set of real numbers.
A random variable is a set of ordered pairs (X, Y) in which X belongs to the set of possible outcomes of a random experiment and Y belongs to the set of probabilities of obtaining any one of these outcomes.
Example
When rolling an honest die with six faces numbered 1 to 6, the possible outcomes belong to the set {1,2,3,4,5,6} and the set of probabilities of the events is a subset of the interval [0, 1]. The value 0 corresponds to an impossible event and the value 1 corresponds to a certain event.
The probability of rolling an even number is 0.5 or \(\frac {1} {2}\).
The random variable f can therefore be described in extension like this:
\(f=\{ \left( 1,\frac {1} {6}\right) ,\left( 2,\frac {1} {6}\right),\left( 3,\frac {1} {6}\right),\left( 4,\frac {1} {6}\right),\left( 5,\frac {1} {6}\right),\left( 6,\frac {1} {6}\right)\}\).