How do I symbolically notate the pdf of a distribution?

Question

For example, given the distribution $\mathcal{D}$, such as the normal distribution $\mathcal{N}(\mu,\sigma)$, how would I write $PDF(\mathcal{D},x)$ without explicitly writing out the function as ${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$?

Something like $\mathcal{N}_x(\mu,\sigma)$? What criteria on $x$ should be specified?

Answer 1

Probability distributions are related to random variables and so the common practice is to denote such density function as $f_X$, if the random variable is denoted by $X$.

Answer 2

The statement

Let $X\sim\mathcal N(\mu,\sigma^2)$ denote a normally distributed random variable with mean $\mu$ and variance $\sigma^2$.

implies that $X$ admits a density and distribution function of the form $$ f_X(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) $$ and $$ F_X(x)={\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}, $$ respectively. There is no need to include an $x$ in the manner you specified, i.e. $\mathcal N_x(\mu,\sigma^2)$.

Note the use of the subscript $X$ in the notation pertaining the density and distribution functions to make it clear that they correspond to the random variable $X$. This is standard notation which can be used for other types of random variables. For example, $X\sim\operatorname{Gamma}(\alpha,\beta)$ denotes a gamma random variable and we can write the corresponding density as $$ f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}. $$