On discrete random variables

The book states that

"a discrete random variable is a random variable with countable number of outcomes.

For example, a discrete random variable X can take on a limited number of outcomes or a discrete random variable Y CAN take on an UNLIMITED number of outcomes. Because we can count all the possible outcomes of X and Y, both X an Y satisfy the definition of a discrete random variable."

How is random variable Y a discrete random variable when it is unlimited?

If it can take on the values 1, 2, 3, 4, . . . (i.e., all of the counting numbers), it is a discrete random variable.

Sorry i still dont get it. My understanding is that random variable Y is unlimited. Sure, it will contain numbers 1,2,3,4…999999 and beyond, but since the possible number of outcomes can’t be countd, it can’t be a discrete random variable.

It’s a discrete random variable if there are gaps between the possible values.

The counting numbers have gaps between them (e.g., between 1 and 2 you have 1.5, √2, π/e, and so on), so a probability distribution on the counting numbers is a discrete distribution.

It’s the gaps that make it a discrete distribution, not whether the outcomes are bounded or not.


You have a misunderstanding of the term “countable”.

Countable does not mean finite. A set is countable if you can put it into a one-to-one correspondence with a subset of the counting numbers (1, 2, . . . .). If that subset has an upper bound, then it is finite; all finite sets are countable. But if it hasn’t an upper bound, it is (countably) infinite; countably infinite sets are countable.

As a couple of examples, the set of all rational numbers (p/q, p & q integers, q ≠ 0) is countable (countably infinite), whereas the set of all real numbers is _ uncountably _ infinite.