# Convention to denote CDF of "named" distribution

I have found the following convention useful in my own work: If a (family of) distribution(s) has a standard symbol, for example the normal distribution $$X \sim \mathcal N (\mu, \sigma^2),$$ then I write its probability density function as $$f(x) = \mathcal N (x ; \mu, \sigma^2).$$ Unfortunately, I don't remember where I picked up this convention.

Given this, is there a standard / common extension to denote the cumulative distribution function? Possibilities that come to mind are an accent, e.g. $$F(x) = \tilde {\mathcal N} (x ; \mu, \sigma^2),$$ or a subscript $$F(x) = \mathcal N_\mathrm{CDF} (x ; \mu, \sigma^2).$$

I realize that this is a "soft" question, but in my experience efficient notation is an important tool in mathematical / statistical work.

• Your notation for $f$ is not a convention: it is ambiguous.
– whuber
May 1 '19 at 21:03
• Another convention I've occasionally seen is $F_\mathscr{N}$.
– Ben
May 2 '19 at 3:36
• @whuber, I don't see how my notation is ambiguous. Care to elaborate? May 4 '19 at 17:08
• @Ben, yes that works. I hesitate to use it because it becomes cumbersome when parameters are specified: $F_{\mathcal N(\mu, \sigma^2)}$. May 4 '19 at 17:10
• Your notation could refer to any description of that distribution. One would guess the intent is either the CDF or the PDF, but the cgf, cf, and other mathematical objections would qualify as well.
– whuber
May 4 '19 at 21:52

No, there's no standard, neither for generic nor for "named" distributions. I have seen a sort of a convention to denote PDF with small cap f and CDF with capital F, e.g. see MathWorks MATLAB docs. Even this is not everywhere, e.g. Wolfram Mathematica uses P and D for these.

The symbol $$\mathcal N$$ usually denotes normal distribution, not its PDF or CDF.

• Yes, $f$ / $F$ is pretty standard, but that only works for generic distributions. My question is about "named" distributions: Normal, binomial, gamma, etc. May 4 '19 at 17:08

For cdf $$F(x) = \Phi(\frac{x-\mu}{\sigma})$$ where $$\Phi$$ is the standard normal cdf;

similarly for pdf, $$f(x) = \frac{1}{\sigma}\phi(\frac{x-\mu}{\sigma})$$ where $$\phi$$ is the standard normal density.

These are commonly used; it works because the normal is a location-scale family.

I don't remember where I picked up this convention.

I've seen some finance people (and perhaps one or two others) write the density the way you do, it's not standard in the stats literature, given there's already a less ambiguous notation in common use.

• Thanks! I'm familiar with the $\phi$ / $\Phi$ notation. However, my question was about a general notation, the normal distribution was – as stated – just an example. May 4 '19 at 17:07
• The usual approach is to name the distribution and denote its cdf symbolically (as a function); my answer reflects that practice on your example. Where there's no conventional symbol for the cdf of a particular distribution, $F$ or $G$ are common choices (following the convention of Roman capitals for cdfs), but many other symbols are used as circumstances demand. However, its important to note that parameterizations can vary so its important to be clear how you're parameterizing the cdf (at least when you move away from the normal). May 5 '19 at 1:20
• Introducing extra symbols is exactly what I'm trying to avoid. And the parametrization can be ambiguous, but that's true for the standard distribution symbols as well. Even for the normal, people disagree whether to use the variance or the standard deviation as a parameter. May 5 '19 at 13:44