Probability Mass Function of Function of Discrete Random Variable

Theorem
Let $X$ be a discrete random variable.

Let $Y = \map g X$, where $g: \R \to \R$ is a real function.

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

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

Thus: