2.5. Random Variables

Remember the random experiment of throwing with a die? We defined events then as “6” or “odd”. The events “1”,…,”6” of course are the elementary events in the universe \(U\) of all possible outcomes. Note that we write “1” here instead of 1 to stress that we are labeling events.

Let \(u\in U\) be such an elementary outcome in the universe \(U\). Then we can set up a mapping \(X: u\in U\mapsto X(u)\in B\) where \(B\) is either \(\setZ\) or \(\setR\). Such a mapping is called a random variable. A RV is a numerical observation of the outcome of a random experiment.

Consider again the experiment of throwing with a fair die. In this case \(u\in\{\text{"1"}, \text{"2"}, \ldots,\text{"6"}\}\). An obvious mapping in this case is \(X(\text{"1"}) = 1\), \(X(\text{"2"}) = 2\) etc. This is an example of a discrete random variable, the range of the mapping \(X\) is the set of integers \(\setZ\).

Now consider the experiment where we measure the length of a randomly selected person, so \(U\) is the set of all persons. The length can be any real number in the range of \(0\) to \(\infty\) (of course not all lengths are equally probable). The random variable in this case is \(X(u) = \text{length of person } u\), i.e. \(X: u\in U \mapsto \setR\). Such a RV is called a continuous random variable.