Probability Mass Function

Definition 11.3 (Probability mass function). If $X$ is a discrete random variable taking the values $x_1,x_2,x_3,\dots,x_n,\dots$, the function $f(\cdot)$ (also written $p(\cdot)$) defined by

is the probability mass function (pmf) of $X$.

Theorem 11.1 (characterisation, without proof). $f(x)$ is a pmf if and only if

(i) $f(x_k)\ge0$ for every $k$, and (ii) $\displaystyle\sum_k f(x_k)=1$.

Worked illustrations. …