ABSTRACT
In the previous chapter we observed that the number of support points for a discrete random variable is either finite or at most countably infinite. Masses assigned to these points add up to one. In the case of a continuous random variable, however, the support contains at least one entire interval on the real line, so that the number of points in the support of such a random variable is necessarily uncountable. Unfortunately, the mathematics does not work when one starts assigning positive probabilities to each individual point in the support in this case, since the total mass (probability) can then no longer be bounded by one. Thus, under continuous models, single points cannot carry any mass. So if X is a continuously distributed random variable, Pr[X = k] is always equal to zero for any fixed value k in the support of the random variable. By the same rule, the probability statements for open and the corresponding closed intervals are always equal for a continuous random variable; for example, Pr[X < k] = Pr[X ≤ k].
