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:

$\displaystyle p_Y \left({y}\right) = \sum_{x \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)$

Proof

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

Thus:

	\(\displaystyle \)	\(\displaystyle p_Y \left({y}\right)\)	\(=\)	\(\displaystyle \Pr \left({Y = y}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Pr \left({g \left({X}\right) = y}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Pr \left({X \in g^{-1} \left({Y}\right)}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{x \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)\)	\(\displaystyle \)

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 2.3 \ (18)$