A FEW USEFUL DISCRETE DISTRIBUTIONS

There is a wide family of discrete probability distributions, which we cannot cover exhaustively. Nevertheless, we may get acquainted with the essential ones, which will be illustrated by a few examples. First, we should draw the line between empirical and theoretical distributions. Since these terms may be a tad misleading, it is important to clarify their meaning.

• An empirical distribution, in the discrete case, consists of a set of values along with their probabilities. Hence, to describe it, we need the full PMF. This is why the term nonparametric distribution is also used.
• A theoretical or parametric distribution stems from a specific random experiment that may be conceptual, rather than empirical. This will be clearer in a moment, but the important point is that such a distribution can be fully characterized by a limited set of parameters, typically one or two. The shape of the distribution depends on the specific mechanism of the underlying random phenomenon.

The terminology for theoretical distributions is as such, because these distribution often allow for an infinite support, something that cannot stem from empirical experiments; an empirical distribution has a finite support by its very nature. Still, the parameters of a theoretical distribution can be fit against empirical data. Hence, we see that the boundary between the two classes is not that sharp, and is more related to their parametric vs. nonparametric nature. In describing the distributions below we will also use a few realistic examples to hone our skills.