Random Variable

Formally, the random variable is defined as a function that maps the outcome space to the real space,

\[ X : \Omega \to \mathbb R \]

Random Variable as a Set

As a set, the random variable the result of mapping the outcome space to the real space \(X = \set{x_1, x_2, \ldots, x_n},~~ x_i \in \mathbb{R}\)

An event occuring in the outcome space \(A \subseteq \Omega\) corresponds to an event occuring in the real space \(A' \subseteq X\) (limited to the subspace of \(X\)) such that,

\[\begin{split} X \cup A' = \set{x : x \in A'}\\ A' = X \cup A' \end{split}\]

We can get some intution for these two concepts with the equation for the norm of a probabilistic being 1:

\[ \sum_{x\in X} P(x) = 1 \tag{2nd Axiom} \]

Random Variable as a Function

\[ X(\omega_i) = x_i \in \mathbb R \]

If an event \(A \subseteq \Omega\) occurs in outcomes space then a corresponding event \(A' \subseteq \mathbb{R}\) occurs in real space such that,

\[\begin{split} X : A \rightarrow A' \equiv \set{X \in A'} = \set{\omega : X(\omega) \in A'}\\ A' = X \in A' \end{split}\]