Absolute continuity of distributions --- why do we need this? [duplicate]

Question

Why do we define continuous probability distribution as those with absolutely continuous CDF, instead of just continuous CDF?

Answer 1

You can find more detail on this issue in a related question here. Absolute continuity of the CDF is a stronger condition than continuity, and essentially just means that the distribution has a valid density function. For example, if the CDF $F$ of a real scalar random variable is absolutely continuous then there exists a real function $f$ (the density) such that:

$$F(x) = \int \limits_{-\infty}^x f(x) dx.$$

This requirement does not hold for all continuous distributions. There are some nasty distributions (such as the notorious Cantor function) that are continuous, but are not absolutely continuous, so they have no density function that satisfies the above equation.

This isn't really a matter of how we define the distributions. It's just that, in practice, all the uniformly continuous distributions we use in statistics are also absolutely continuous ---i.e., they have valid density functions. You will occasionally encounter cases of random variables that are a mixture of continuous and discrete parts, but practical applications do not give rise to the dread cases where you have a continuous, but not absolutely continuous, distribution.