Which statement describes discrete random variables?

Which statement describes discrete random variables?

• A. Discrete random variables have countable outcomes.
• B. Discrete random variables have uncountable outcomes.
• C. Both discrete and continuous have countable outcomes.
• D. Neither has countable outcomes.

Answer: (A)

Discrete random variables are defined by outcomes that you can list one by one, either a finite set or a countably infinite set. That means the possible values are countable. For example, the result of rolling a six-sided die has only six outcomes, while the number of customers arriving in an hour can be any nonnegative integer, which is still countable. In contrast, continuous random variables take values from intervals of real numbers and have uncountably many possible values. Heights, times, or temperatures can vary smoothly and can take infinitely many values within a range, so you can’t list them all one by one. Probabilities in this case are described over ranges (via a density) rather than for individual points. So the statement that discrete random variables have countable outcomes is correct because it captures the essential distinction: their possible values are either finite or countably infinite, unlike continuous variables which have uncountable outcomes.

Discrete random variables are defined by outcomes that you can list one by one, either a finite set or a countably infinite set. That means the possible values are countable. For example, the result of rolling a six-sided die has only six outcomes, while the number of customers arriving in an hour can be any nonnegative integer, which is still countable.

In contrast, continuous random variables take values from intervals of real numbers and have uncountably many possible values. Heights, times, or temperatures can vary smoothly and can take infinitely many values within a range, so you can’t list them all one by one. Probabilities in this case are described over ranges (via a density) rather than for individual points.

So the statement that discrete random variables have countable outcomes is correct because it captures the essential distinction: their possible values are either finite or countably infinite, unlike continuous variables which have uncountable outcomes.