When reading mathematical texts, I notice terminology such as the following:
Let $X$ be a random variable with probability density function (PDF) $f(x) = \dots$
Note: the dummy variable $x$ in the PDF definition shouldn't be confused with the random variable $X$ here. Or sometimes, when the PDF is unknown, it is referred to like this:
Let $X$ be a random variable with some probability density function (PDF) $f_X$
Clearly, the function $f$ or $f_X$ can be uniquely determined from the random variable $X$ and every single random variable that we care to define has such an associated PDF, by definition. So it's a bit cumbersome to have to write the English sentence linking the PDF to the random variable. Instead, is there a universally accepted notation that unambiguously refers to the PDF associated to a random variable $X$, given just the name of the random variable?
Example Usage
For example, let's say the standard accepted notation for the above was $\Theta_X$. I'm not saying it is, but just assume it is for the sake of this question so you can see the point of what I'm asking. We could then write something like this:
Suppose we have a random variable $X$. Then for any $a, b \in \mathbb{R}$, we have: $$P(a \leq X \leq b) = \int_a^b\Theta_X(x)dx \leq 1$$
- Notice that I don't need to define $\Theta_X$ as the associated PDF to $X$ in the above, because I've globally assumed it as accepted notation.
- Also notice that I used the notation $P(a \leq X \leq b)$ here, because that seems to be globally accepted to mean the probability that the random variable $X$ is between $a$ and $b$.
- It might seem lazy in this case to not just explicitly say $\Phi_X$ is the associated PDF to $X$ when defining $X$, but when we have mathematical texts with many random variables floating around it starts to get cumbersome.
A Note on Tilde
I have seen $\sim$ used to link a random variable to a PDF in some cases, particularly when the PDF is normal, but this is still a bit clunky as we have to elsewhere show the relationship. For example, on Wikipedia's Normal Distribution page we have:
$X \sim \mathcal{N}(\mu, \sigma^2)$
To mean $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$.
Best Answer
Mathematical writing should be all about clarity, not brevity. "Suppose the CRV $X$ has PDF $f_X$" is far clearer than the idea you suggested, and should therefore be preferred.