# Probability Mass Function of Function of Discrete Random Variable

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## 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:

 $\ds \map {p_Y} y$ $=$ $\ds \map \Pr {Y = y}$ $\ds$ $=$ $\ds \map \Pr {\map g X = y}$ $\ds$ $=$ $\ds \map \Pr {X \in \map {g^{-1} } Y}$ $\ds$ $=$ $\ds \sum_{x \mathop \in \map {g^{-1} } y} \map \Pr {X = x}$

$\blacksquare$