Probability Mass Function of Function of Discrete Random Variable

Theorem
Let $X$ be a discrete random variable.

Let $Y = g \left({X}\right)$, where $g: \R \to \R$ is a real function.

Then the probability mass function of $Y$ is given by:
 * $\displaystyle p_Y \left({y}\right) = \sum_{x \mathop \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)$

Proof
By Function of Discrete Random Variable‎ we have that $Y$ is itself a discrete random variable.

Thus: