\begin{definition}[Definition:Independent Random Variables]
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ and $Y$ are defined as '''independent (of each other)''' {{iff}}:
:$\map \Pr {X = x, Y = y} = \map \Pr {X = x} \map \Pr {Y = y}$
where $\map \Pr {X = x, Y = y}$ is the joint probability mass function of $X$ and $Y$.
Alternatively, this condition can be expressed as:
:$\map {p_{X, Y} } {x, y} = \map {p_X} x \map {p_Y} y$
Using the definition of marginal probability mass function, it can also be expressed as:
:$\ds \forall x, y \in \R: \map {p_{X, Y} } {x, y} = \paren {\sum_x p_{X, Y} \tuple {x, y} } \paren {\sum_y p_{X, Y} \tuple {x, y} }$
\end{definition}