> [!NOTE] Definition > A [[Function|function]] $f:\mathbb{R}\to [0,1]$ is a probability mass function iff its [[Support of a real-valued function|support]], given by $D=\{ x\in \mathbb{R}: f(x)>0 \},$ is [[Countable Set|countable]] and $\sum_{x\in D} f(x)=1.$ # Properties By [[Probability Mass Function Defines Discrete Real-Valued Random Variable]], if $f$ is a PMF, then there exists a probability space and a discrete real-valued random variable on the this space such that its PMF is given by $f$: that is $\mathbb{P}(X^{-1}(x))=f(x),$ for all $x\in \mathbb{R}.$