Chapter 1 enabled us to familiarize ourselves (say to revisit, or to brush up?) the concept of probability. As we have seen, any probability is associated with a given event xi from a given event space S = {x1, x2,…, xN}. The discrete set {p(x1), p(x2),…, p(xN)} represents the probability distribution function or PDF, which will be the focus of this second chapter.

So far, we have considered single events that can be numbered. These are called discrete events, which correspond to event spaces having a finite size N (no matter how big N may be!). At this stage, we are ready to expand our perspective in order to consider event spaces having unbounded or infinite sizes (N → ∞). In this case, we can still allocate an integer number to each discrete event, while the PDF, p(xi), remains a function of the discrete variable xi. But we can conceive as well that the event corresponds to a real number, for instance, in the physical measurement of a quantity, such as length, angle, speed, or mass. This is another infinity of events that can be tagged by a real number x. In this case, the PDF, p(x), is a function of the continuous variable x.

This chapter is an opportunity to look at the properties of both discrete and continuous PDFs, as well as to acquire a wealth of new conceptual tools!