Expectation of Function of Discrete Random Variable

Theorem
Let $X$ be a discrete random variable.

Let $E \left({X}\right)$ be the expectation of $X$.

Let $g: \R \to \R$ be a real function.

Then:
 * $\displaystyle E \left({g \left({X}\right)}\right) = \sum_{x \in \Omega_X} g \left({x}\right) \Pr \left({X = x}\right)$

whenever the sum is absolutely convergent.

Proof
Let $\Omega_X = \operatorname{Im} \left({X}\right) = I$.

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

Thus $\Omega_Y = \operatorname{Im} \left({Y}\right) = g \left({I}\right)$.

So:

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