Properties of a discrete random variable My stats course just taught me that a discrete random variable has a finite number of options ... I hadn't realized that.  I would have thought, like a set of integers, it could be infinite. Googling and checking a several web pages, including a few from university courses, has failed to specifically confirm this; most sites do however say discrete random variables are countable - I suppose that means finitely numbered?
It is clear that continuous random variables are infinite even if (most?) often bounded.
But if discrete random variables have finite possibilities, what then is an infinite distribution of integers?  It's neither discrete nor continuous?
Is the question moot because variables either tend to be continuous & (by definition) infinite or discontinuous & finite?
 A: I'm writing an answer, with the perspective that I have only a very naive understanding of measure-theoretic probability (so, experts, please correct me!).
A (real-valued) random variable is a function $X: S \to \mathbb{R}$, where $S$ is a sample space.
$X$ is discrete if $X(S)$, the image of $S$ induced by $X$, is countable. $X$ is continuous if $X$ has an absolutely continuous CDF. (I don't know that much about absolutely continuous functions, so I can't elaborate on this point.)
However, not all random variables are only discrete or continuous. There are "mixed" random variables, where $X(s)$ has a CDF which is the sum of a step function and a continuous function with indicators.
You can also have random variables which are neither discrete nor continuous, such as the Cantor distribution.
A: To quote the wikipedia page on continuous and discrete variables:

If it [the variable] can take on two particular real values such that it can also
  take on all real values between them (even values that are arbitrarily
  close together), the variable is continuous in that interval

Therefore, a discrete random variable does not have to have a 'finite number of options', but there needs to be a non-infinitesimal gap between the possible values. This is the case with a distribution of integers, since the 'distance' between two neighboring integers is 1 and cannot be less than this. Therefore the variable is not continuous as it does not 'continue' within these gaps.
Edit: I know that there's probably better and/or more precise ways of answering this, but this is what helped me personally understand the difference.
A: If that's what your course said, it's wrong.
While discrete distributions can have a finite number of possible outcomes, they are not required to; you can have a discrete distribution that has an infinite number of possible outcomes - the number of elements should be no more than countable. 
A common example would be a geometric distribution; consider the number of tosses of a fair coin until you get a head. There's no finite upper bound on the number of tosses that may be needed. It may take 1 toss, or 2, or 3, or 100, or any other number. 
A discrete distribution could be negative (consider the difference between two such geometrically-distributed random variables; it can be any positive or negative integer).
A discrete distribution doesn't have to be over the integers, though, like in my example. That's just a common situation, not a requirement. 
