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:

\(\ds \map {p_Y} y\)

\(=\)

\(\ds \map \Pr {Y = y}\)

\(\ds \)

\(=\)

\(\ds \map \Pr {\map g X = y}\)

\(\ds \)

\(=\)

\(\ds \map \Pr {X \in \map {g^{-1} } Y}\)

\(\ds \)

\(=\)

\(\ds \sum_{x \mathop \in \map {g^{-1} } y} \map \Pr {X = x}\)

$\blacksquare$

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.3$: Functions of discrete random variables: $(18)$