Random Variable

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

\[\begin{split} X : \Omega \to \mathbb R\\ \big\downarrow \\ X(\omega_i) = x_i \in \mathbb R \end{split}\]

• We can further define as outcome set in real space as \(\Omega_X\) which is the result of,

  \[\begin{split} \Omega \stackrel{X}{\longrightarrow} \Omega_X\\ \Omega_X \equiv \set{x: X(\omega \in \Omega)} \end{split}\]

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

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

Distribution

The random variable carries the information of the probability distrubtion such that there always exist the probability function for all \(x \in \Omega_X\) denoted as:

\[ P_X(x) \text{ or } P(X=x) \]

Equivalence of Two Random Variables

Two random variables \(X\) and \(Y\) are equivalent only if they represent the same space in the outcome space. However in most cases, \(X\) and \(Y\) represent two different trials (e.g., first flip and second flip). In this case the two random variables can only be equal in distribution denoted as,

\[\begin{split} X \stackrel{d}{=} Y\\ X \ne Y \end{split}\]