Given a discrete
random variable 5.2.1 \(X\) on a space
\(R\text{,}\) a probability mass function on
\(X\) is given by a function
\(f:R \rightarrow \mathbb{R}\) such that:
\begin{align*}
& \forall x \in R , f(x) \gt 0\\
& \sum_{x \in R} f(x) = 1\\
& A \subset R \Rightarrow P(X \in A) = \sum_{x \in A}f(x)
\end{align*}
For \(x \not\in R\text{,}\) you can use the convention \(f(x)=0\text{.}\)