Definition of discrete random variable

When an experiment that has a random outcome is carried out, we are looking out for all the possible outcomes – these are usually parameterized in such a way that an element of a set (e.g., a number) is associated to each outcome. More precisely, to each event \(E\) of a given sample space \(S\) we assign a number \(x\). This then gives rise to a function \(X\). We define this function \(X\) to be a random variable.

We take the usual notational practice adopted by probability theorists, i.e., random variables ranges over uppercase letters, e.g., \(X, Y, Z\), and the values of the corresponding random variables over lower case letters, e.g., \(x, y, z\).

For a given random variable \(X\) and a particular value of it \(x\), we form the instance \(X = x\) which represents the event we are interested in calculating its probability \(P(X = x)\). We use the word probability distribution of \(X\) to talk about the values of \(P(X = x)\) as \(x\) ranges over all possible instances in the event space. We often write

\[\mathrm{f}_X(x) = P(X = x),\]

and we call \(\mathrm{f}_x\) the probability density function of \(X\). Summing up over all possible \(x\) in the event space, we obtain

\[\sum_{\text{all}\hspace{0.5em} x} P(X = x) = \(1\).\]