On discrete random variables

[Postante]

I have seen the following "extension" of discrete random variables definition, from:
pediaview.com/openpedia/Probability_distributions
(Abstract)
"... Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. The points where jumps occur are precisely the values which the random variable may take. The number of such jumps may be finite or countably infinite. The set of locations of such jumps need not be topologically discrete; for example, the cdf might jump at each rational number."
Do you agree with the statement that the cdf of a DRV jumps at each rational number?

 

[Ayre]

I'm inclined to agree, i.e. I think that a random variable whose cdf jumps at all the rationals can be called discrete.

The rationale? Well, the concept of random variable doesn't have anything to do with the topology of the space of values of rational numbers. It would be silly to require that the concept of a random variable being discrete depend on the topology of the value space.

And if you don't consider the topology, there's nothing to distinguish two random variables taking values in ℤ and a random variable taking values in ℚ. As the former is called discrete, so should the latter be.

 

[Postante]

The problem is as it follows:
"In cases more frequently considered, this set of possible values is a topologically discrete set in the sense that all its points are isolated points. But there are discrete random variables for which this countable set is dense on the real line (for example, a distribution over rational numbers)."
This means that the jumps are dense on the real line, like those of Cantor's function.
I believe that such a "discrete" rv actually is a continuous, but not "absolute continuous", random variable. It is a singular distribution (Lukacs: "Characteristic Functions", Griffin, 2nd Ed, 1970).