I defined it as follows:
Let us consider a random variable $X$ defined on a countable sample space $\Omega$ over $K$ possible outcomes, and $P(X)$ its discrete PD defined as the set of probabilities that $X$ takes on the non-zero probability values $x_{i}$ $(i=1,..,C)$ as: \begin{equation} P(X=x_{i})=p_{i}=\frac{n_{i}}{N}, \label{Prob} \end{equation} with $n_{i}$ the number of events relative to the i-th possible outcome, $N$ the overall number of events or measurements, and $C$ the number of all possible outcomes, such that, in the limit of the large numbers ($N \rightarrow \infty$), for a normalized PD, $\sum_{i} p_{i}=1$.
What is wrong with the above definition? I was reprimanded for unclear mathematical language. For example, I didn't specify that random variables are measurable functions from the sample space into K. Well, ok, but isn't a random variable a measurable function defined over a (here countable) sample space? Just wondering how to rephrase the whole thing in a correct and/or more rigorous manner?