Expectation of Function of Joint Probability Mass Distribution

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

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

Let $$E \left({X}\right)$$ 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 $$Y$$.


 * $$E \left({g \left({X, Y}\right)}\right) = \sum_{x \in \Omega_X} \sum_{y \in \Omega_Y} g \left({x, y}\right) p_{X, Y} \left({x, y}\right)$$

whenever the sum is absolutely convergent.

Proof
Let $$\Omega_X = \operatorname{Im} \left({X}\right) = I_X$$ and $$\Omega_Y = \operatorname{Im} \left({Y}\right) = I_Y$$.

Let $$Z = g \left({X, Y}\right)$$.

Thus $$\Omega_Z = \operatorname{Im} \left({Z}\right) = g \left({I_X, I_Y}\right)$$.

So:

$$ $$ $$ $$

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