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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 $\sigma^2$.

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    $\begingroup$ I''d advise against $\Phi$, because it's a CDF symbol for $\mathcal{N}(0,\,1)$. $\endgroup$
    – J.G.
    Jun 6, 2020 at 12:13
  • $\begingroup$ As the answers mention, a random variable doesn't always have a PDF. However, we can define the (cumulative) distribution function for any random variable and in the following video, around 5:15, the lecturer uses the notation $F_X$ to denote the distribution function associated to $X$. youtube.com/… $\endgroup$ Oct 29, 2020 at 18:27

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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.

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    $\begingroup$ Agreed. I don't want to make this an opinion-baesd post though please - just want to check if there is such a global symbol that I'm missing out on. Thanks. I'm assuming there isn't, and I'm happy to go on typing out the PDFs explicitly if so. $\endgroup$ Jun 6, 2020 at 11:39
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    $\begingroup$ As you mentioned, I think the tilde is the only widely used symbol for this. $\endgroup$
    – K.defaoite
    Jun 6, 2020 at 11:40
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    $\begingroup$ @ColinBhandal As I've mentioned before, if you can't find a symbol at first, people not might understand one you do find. Note that $X\sim$ should be followed by a distribution, not that distribution's PDF. $\endgroup$
    – J.G.
    Jun 6, 2020 at 12:15
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The short answer is 'no', so I recommend being clear as suggested in another answer.

A couple of things though:

You say "every single random variable that we care to define has such an associated PDF". This is not true. It is not the case that every random variable has a pdf or every 'nice' random variable as a pdf.

You also say ''I have seen ∼ used to link a random variable to a PDF in some cases''. This seems to confuse the distribution of a random variable with its probability density function.

I think it's an honest question you are asking but there are some conflations going on in your understanding which need straightening out before you can make sense of all of this.

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  • $\begingroup$ True,true. Would you mind linking me to some resources clarifying the subtleties you allude to and hopefully I can edit the question statement to be more correct when my understanding grows. $\endgroup$ Jun 6, 2020 at 13:14
  • $\begingroup$ Also funny is that any random variable has a density with respect to the measure induced by the CDF. But we understand density with respect to the Lebesgue measure and that indeed does not always exist. $\endgroup$
    – Shashi
    Jun 6, 2020 at 13:37
  • $\begingroup$ Thanks Shashi. I will probably need to look into measure theory more deeply. Will hopefully update the question at some point. $\endgroup$ Jun 6, 2020 at 13:50

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