Condition for Independence of Discrete Random Variables

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ and $Y$ are independent there exist functions $f, g: \R \to \R$ such that the joint mass function of $X$ and $Y$ satisfies:
 * $\forall x, y \in \R: \map {p_{X, Y} } {x, y} = \map f x \map g y$

Proof
We have by definition of joint mass function that:
 * $x \notin \Omega_X \implies \map {p_{X, Y} } {x, y} = 0$
 * $y \notin \Omega_Y \implies \map {p_{X, Y} } {x, y} = 0$

Hence we only need to worry about values of $x$ and $y$ in their appropriate $\Omega$ spaces.

Necessary Condition
Suppose there exist functions $f, g: \R \to \R$ such that:
 * $\forall x, y \in \R: \map {p_{X, Y} } {x, y} = \map f x \map g y$

Then by definition of marginal probability mass function:
 * $\displaystyle \map {p_X} x = \map f x \sum_y \map g y$
 * $\displaystyle \map {p_Y} y = \map g y \sum_x \map f x$

Hence:

So it follows that:

Hence the result from the definition of independent random variables.

Sufficient Condition
Suppose that $X$ and $Y$ are independent.

Then we can take the variables:
 * $\map f x = \map {p_X} x$
 * $\map g y = \map {p_Y} y$

and the result follows by definition of independence.