The domain of continuous random variables

Question

This is the definition of random variables I found out:

A random variable is a function that assigns values to each of an experiment's outcomes.

For example, if we toss two different coins at a time and let random variable $X$ be the numbers that the coins come up heads, then we have $X : \{HH, HT, TH, TT\} \rightarrow \{0,1,2\}$ where, for example, $X(HH) = 2.$

However, a problem occurs when I think about continuous random variables. For example, if I let $X$ be height of students. I understand that random variable $X$ is continuous since its value can be any real numbers in, let's say, an interval $[140, 200].$ However, what is the domain of this random variable $X$? If the domain is the set of students, then the range can be listed.

Answer 1

In the continuous case you mention we can choose $X$ to be the function $[140,200]\to\mathbb R$ that is prescribed by $\omega\mapsto\omega$.

That is not a "must" however. We are dealing with a real world situation that must be modeled by means of a probability space and there are lots of ways to do that.

The one I just prescribed is the most natural one (i.e. the first thing that comes up in my mind).

A list of students will not do here as outcome space, because then automatically $X$ is a function that can only take a finite number of values.

Answer 2

From Wikipedia:

The domain of a random variable is a sample space, which is interpreted as the set of possible outcomes of a random phenomenon.

As you can see, in the case of students' heights, the sample space and range are the same set. Because the outcome is the same as the numerical value, if that makes sense. For a coin flip, heads/tails needs to be transformed into $\pm 1$.