Question/Request: Can you please verify if the following analogy would be a good way to understand/introduce multiple random variables? There may be some caveats but if it is getting to the intuitive feel of the actual idea then I would be happy with that.
I'm trying to draw an analogy between a multiple random variables and "parametric equations" as taught in high school. Would this be a good way to think about it?
The definition of a random variable for a sample space $S$ with outcomes $\omega$ is $$\omega\to (X(\omega), Y(\omega))$$
which is quite similar to a parametric equation $$t\to(x(t), y(t))$$
Is the reason why we need the joint distribution (maybe analogous to the "cartesian equation" in parametrics) so we can accurately capture how $X$ and $Y$ move together, instead of focusing on them individually?
The point where my analogy might not work/be accurate: The way two individual marginal distributions of $X$ and $Y $ is not enough to know the joint distribution is similar to how it's not always possible to "eliminate the parameter" in parametric equations to get the cartesian equation. To push it even further, when your random variables are independent, you can get the joint distribution just from the marginal distributions, similar to when you are able to find the cartesian equation from the two individual parametric equations $x(t)$ and $y(t)$ by eliminating the parameter.