Definition:Independent Random Variables

Definition
Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $X$ and $Y$ be random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Then $X$ and $Y$ are defined as independent (of each other) iff:
 * $\Pr \left({X = x, Y = y}\right) = \Pr \left({X = x}\right) \Pr \left({Y = y}\right)$

where $\Pr \left({X = x, Y = y}\right)$ is the joint probability mass function of $X$ and $Y$.

Alternatively, this condition can be expressed as:
 * $p_{X, Y} \left({x, y}\right) = p_X \left({x}\right) p_Y \left({y}\right)$

Using the definition of marginal probability mass function, it can also be expressed as:
 * $\displaystyle \forall x, y \in \R: p_{X, Y} \left({x, y}\right) = \left({\sum_x p_{X, Y} \left({x, y}\right)}\right) \left({\sum_y p_{X, Y} \left({x, y}\right)}\right)$

General Definition
The definition can be made to apply to more than just two events.