Expectation of Function of Joint Probability Mass Distribution

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

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

Let $\expect X$ be the expectation of $X$.

Let $g: \R^2 \to \R$ be a real-valued function

Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$.


 * $\ds \expect {\map g {X, Y} } = \sum_{x \mathop \in \Omega_X} \sum_{y \mathop \in \Omega_Y} \map g {x, y} \map {p_{X, Y} } {x, y}$

whenever the sum is absolutely convergent.

Proof
Let $\Omega_X = \Img X = I_X$ and $\Omega_Y = \Img Y = I_Y$.

Let $Z = \map g {X, Y}$.

Thus $\Omega_Z = \Img Z = g \sqbrk {I_X, I_Y}$.

So:

From the definition of expectation, this last sum applies only when the last sum is absolutely convergent.