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Suppose I want to define some probability distribution in a paper, and exhibit its PDF. It seems the usual way to do this requires two separate sentences, one to define the distribution and one to define its PDF. For example, one would write:

We will show that this quantity is blah distributed. We say a random variable $X$ is blah distributed if $X \sim \text{Blah}(\alpha, \beta)$. Then the probability density function of such a random variable $X$ is $$f(x; \alpha, \beta) = \beta I \alpha h \times \ldots.$$

This notation is unsuitable to me for two reasons.

  • It's clunky. I only ever need the PDF, so I want a notation that avoids taking an extra step to initialize an object that won't ever be used. (I'm working in a non-stats, non-math field where everybody knows what a PDF is, but nobody ever uses the word 'random variable'. The general reaction to the above phrasing would be that I'm just being pedantic for no reason.)
  • It's missing information. Somebody just skimming through, looking at just the equations, has no idea what $f$ is. The notation has no way to indicate which PDF it is.

I tried to fix these two issues by combining these into a single line:

We will show that this quantity is blah distributed, i.e. its probability density function is $$\text{Blah}(\alpha, \beta)(x) = \beta I \alpha h \times \ldots.$$

But this feels weird. Is this okay? Is there a better way to do this?

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    $\begingroup$ "I'm working in a non-stats, non-math field where everybody knows what a PDF is, but nobody ever uses the word 'random variable'." — Sounds like a very odd field. $\endgroup$ – Kodiologist Aug 21 '16 at 3:37
  • $\begingroup$ @Kodiologist This happens a lot in science! Everybody uses integrals. Nobody ever specifies that their functions are (Riemann, Lebesgue, etc.) integrable. $\endgroup$ – knzhou Aug 21 '16 at 3:38
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With these desiderata (mention the name of the distribution (Blah), give an explicit form of the pdf, do not waste space with $X\sim \mathrm{Blah}(\cdot)$), I would use the density function notation style discussed in this question Meaning of syntax $N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ (multivariate normal distribution), so

\begin{equation} p(x ; \alpha, \beta ) = \mathrm{Blah}(x;\alpha,\beta) = \beta\,I\,\alpha\,h\times\ldots \end{equation}

where the first equation tells the brand name of the distribution and the second specifies the explicit functional form.

However, no guarantees that your readers understand this notation. It would be better to simply check what kind of notations are used in your field.

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