I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and cdf"
Suppose I say: Let $X$ be a random variable with Gaussian CDF.
What does that mean exactly? $X$ is a function that maps from the event space to a real number. What does it mean to "connect" it, "link" it, "equip" it, WITH a CDF? It is already a function, what does it mean to let it hook up with another function, why don't we just deal with $X$ directly.
Then I look at the CDF:
$f_X(x) = \int_A \exp(-x^2/2)dx$
I ask myself: Is "$\exp(-x^2/2)$" part the random variable? No. Is "$\int_A \exp(-x^2/2)dx$" the random variable? No. Is little $x$ is the random variable? No. What is the difference if I wrote $f(x)$ instead of $f_X(x)$? Nothing happens.
What does the CDF/PDF have to do with the random variable exactly? How do you know which CDF/PDF a random variable "has"?