For a discrete random variable $X$ taking the values $x_1,x_2,x_3,\dots$, the probability mass function (pmf), written $f(\cdot)$ or $p(\cdot)$, is defined by

(Definition 11.3). By Theorem 11.1, a function $f$ is a valid pmf exactly when (i) $f(x_k)\ge0$ for every $k$, and (ii) $\sum_k f(x_k)=1$ — every candidate pmf in this chapter is checked against these two conditions, either to find an unknown constant (e.g. solving for $k$ in $f(x)=kx^2$) or simply to confirm the given table is legitimate.

The pmf is the discrete analogue of a "how likely is each value" table, and can be presented in three equivalent forms: as a table ($x$ against $f(x)$), as a graph (a bar/spike diagram over the support points), or as an algebraic expression $f(x)=\dots$ valid over the support. …