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 if and only if 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:
- $\ds \map {p_X} x = \map f x \sum_y \map g y$
- $\ds \map {p_Y} y = \map g y \sum_x \map f x$
Hence:
\(\ds 1\) | \(=\) | \(\ds \sum_{x, y} \map {p_{X, Y} } {x, y}\) | Definition of Joint Probability Mass Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{x, y} \map f x \map g y\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_x \map f x \sum_y \map g y\) |
So it follows that:
\(\ds \map {p_{X, Y} } {x, y}\) | \(=\) | \(\ds \map f x \map g y\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x \map g y \sum_x \map f x \sum_y \map g y\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \sum_y \map g y} \paren {\map g y \sum_x \map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {p_X} x \map {p_Y} y\) | from above |
Hence the result from the definition of independent random variables.
$\blacksquare$
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.
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 3.3$: Independence of discrete random variables: Theorem $3 \ \text{B}$