Let \(X\) be a random variable of a given random experiment. Assume, we have a random experiment, for which the probability of the event.
"\(X\) has a realization equal a given real number \(x\)",
i.e. the probability \(p(X = x)\) exists for all real numbers \(x\in\mathbb R\).
Then we call the function. \[f:=\cases{\mathbb R\mapsto[0,1]\\x\mapsto p(X=x)}\quad\quad\text{for all }x\in\mathbb R\]
the probability mass function (or pmf) of the random variable \(X\).
Proofs: 1 2 3 4
Propositions: 5 6 7