Definition 5.2.1. Random Variable.
Given a random experiment with sample space \(S\text{,}\) a function \(X\) mapping each element of \(S\) to a unique real number is called a random variable. For each element s from the sample space \(S\text{,}\) denote this function by
\begin{equation*}
X(s) = x
\end{equation*}
and let \(R\) be the range of \(X.\)
\(R\) will be called "the space of X" and in notation
\begin{equation*}
R = { x : X(s)=x, \text{for some} s \in S } .
\end{equation*}