About the definition of discrete random variable

[Postante]

About the definition of "discrete random variable"

Hogg and Craig stated that a discrete random variable takes on at most a finite number of values in every finite interval (“Introduction to Mathematical Statistics”, McMillan 3rd Ed, 1970, page 22).
This is in contrast with the assumption that discrete data can take on values that are countably infinite, in particular rational numbers (D.W. Gooch: “Encyclopedic Dictionary of Polymers”, App. E, page 980, Springer, 2nd Ed, 2010).
I would like to know if discrete random variables can – or can not – take on cuontably infinite values in a finite interval. Or, in other words, if the set of possible values of a discrete random variable may be the set of rational numbers.

 

[Simon Bridge]

A DRV must be finite. You need to check the difference between a DRV and discrete data.

Notice - the set of rational numbers would be continuous rather than discrete but you can have discrete data that takes rational-number values.
However, you will never have an infinite number of those data points. The data set takes its values from a countably infinite set, it is not itself countably infinite.
A DRV may draw it's values from a set of discrete data. ie. it will always get it's values from a finite set.

To understand this - consider what sort of process generates discrete random data.
Also see:
http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm

 

[haruspex]

A DRV must be finite. You need to check the difference between a DRV and discrete data.

Notice - the set of rational numbers would be continuous rather than discrete but you can have discrete data that takes rational-number values.
However, you will never have an infinite number of those data points. The data set _takes its values_ from a countably infinite set, it is not itself countably infinite.
A DRV may draw it's values from a set of discrete data. ie. it will always get it's values from a finite set.

To understand this - consider what sort of process generates discrete random data.
Also see:
http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm

I believe a RV taking only rational values would be discrete. I see the following quoted from Valerie J. Easton and John H. McColl's Statistics Glossary v1.1:
"A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values."
A RV with countably many possible values must have defined probabilities at those values.

I don't see the relevance of your discussion of data sets to the OP.

 

[Simon Bridge]

Sure an RV taking rational values can be discrete.
OP brought up data sets in the original question with the reference to Gooch) though not in so many words. I'm not terribly happy with my attempt at a clarification of how Gooch and Hogg-n-Craig are not in conflict.

I like:
"A RV with countably many possible values must have defined probabilities at those values."
I had wondered if I should have included something like that - perhaps as a question:
... if a RV could take any rational number value in [0..1] then what would be the probability of getting a 0.5?

I suppose a better way to think of a DRV is that a probability can be assigned to particular outcomes.

 

[Postante]

An abstract from:
"pediaview.com/Probability_distributions"

... 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.

This means that the definition of DRV is not at all obvious!