# How do I symbolically notate the pdf of a distribution?

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?

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 $$X$$.
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 $$f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}.$$