Definition: Probability Mass Function

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

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  1. Hedderich, J.;Sachs, L.: "Angewandte Statistik", Springer Gabler, 2012, Vol .14