The way I like to think of it is that it is a function that, in a sense, relieves the problem of dealing with nonnumerical elements by assigning each of them a real number (or real-valued vector) so that they can be compared on the real number line.
For instance, let's say I want to figure out how likely it is for a randomly considered member of the population to be no taller than my bio teacher Jim is. Even though it is not impossible to assign a set whose members fit this criteria working entirely within the sample space of human beings, and then to assign a certain measure of probability $P$ to that set, assigning to each person their height is a function that makes this task a bit easier.
Numerically speaking, let $X$ be the function from the sample space of human beings to the (nonnegative) real line that assigns to each person $\omega$ a height $X(\omega)$. Let's say Jim's height is 6 (feet). When we say, then, in layman's terms, what percentile Jim's height represents, what we mean to do is to ask for $P\{\omega : X(\omega) \leq X(Jim)\}$, which is the measure of the set of people (i.e. the probability measuring of the set of people) whose heights do not exceed Jim's. Notice that the function sends us to a nice, ordered place where just considering the people themselves without this numerical value would not be sufficient for this task.
I am not entirely convinced with the line the sample space is also called the support of a random variable
That looks quite wrong to me.
What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $Pr$?
In rather informal terms, the "support" of a random variable $X$ is defined as the support (in the function sense) of the density function $f_X(x)$.
I say, in rather informal terms, because the density function is a quite intuitive and practical concept for dealing with probabilities, but no so much when speaking of probability in general and formal terms. For one thing, it's not a proper function for "discrete distributions" (again, a practical but loose concept).
In more formal/strict terms, the comment of Stefan fits the bill.
Do we interpret the support to be
- the set of outcomes in Ω which have a non-zero probability,
- the set of values that X can take with non-zero probability?
Neither, actually. Consider a random variable that has a uniform density in $[0,1]$, with $\Omega = \mathbb{R}$.
Then the support is the full interval $[0,1]$ - which is a subset of $\Omega$. But, then, of course, say $x=1/2$ belongs to the support. But the probability that $X$ takes this value is zero.
Best Answer
First of all, a random variable is usually defined as a function $X: \Omega \to \mathbb{R}$. So for any possible event in the state space $\omega \in \Omega$, the random variable $X(\omega)$ assigns a real number to that event.
Strictly speaking, probabilities are defined for special sets of events in $\Omega$. They are not defined on the target space $\mathbb{R}$. So if we're being precise, it doesn't makes sense to ask "what is the probability of $X = 3$?" Instead, we should be asking "what is the probability of the set of events corresponding to $X=3$?"
But, there's a catch. Random variables are not just any old functions. They are measurable functions from $\Omega \to \mathbb{R}$. This means that any set of values in the target space $\mathbb{R}$ corresponds to some set of events in $\Omega$, for which a probability has been defined. Therefore, because of this fact we can cut corners and refer to "the probability of $X = 3$," even though it doesn't exactly make sense.
Which brings us to your question. When we speak of "Normal" or "Cauchy" random variables, we are describing how the random variables assign probabilities to events in the state space $\Omega$. We are not actually describing the state space itself. When we say that, for a Normal r.v. $X$ that $P(X \leq 1) = 0.84$, we are really saying that "the probability of all events $\omega$ that $X$ maps to a real number $\leq$ 1 is equal to 0.84." But these events themselves can be anything.
So in short, the answer is: it depends.