Good choice of notation for normal distribution pdf/cdf

Question

Let $\Phi(x)$ be the cumulative distribution function for the standard normal distribution. I have an expression with several terms of the form

$$\Phi\left(\frac{\mu-g(x)}{\sqrt{\sigma^2}}\right)$$ and $$\Phi\left(\frac{\mu+h(x)}{\sqrt{\sigma^2}}\right)$$

where both the expressions for $\mu$, $g(x)$ and $h(x)$ are fairly complicated.

Is there a different way to write these so that the expressions are not so complicated? What I would like is to e.g. write

$$\Phi\left(-g(x);\mu,\sigma^2\right) $$ and $$\Phi\left(h(x);\mu,\sigma^2\right) $$ but I am not sure that it's in good style, since the $\Phi$ denotes the standard normal distribution.

Another way to do it is of course to use

$$F(x;\mu,\sigma^2)=\int_{-\infty}^x\frac{\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right)}{\sqrt{2\pi\sigma^2}}dy$$

but I guess my question is, does there exist some convention for this?

Answer 1

The established convention is for the standard normal distribution's pdf and cdf. But you are allowed to define your own symbols, "for convenience", "for clarity", "for compactness", as usual justifications go.

In you case you could for example define

$$\Phi_{\mu, \sigma} (x)\equiv \int_{-\infty}^x\frac{\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right)}{\sqrt{2\pi\sigma^2}}dy$$

and then use the left-hand side symbol. This up to a degree aligns with the convention that $\Phi_2(x,y;\rho)$ denotes the bivariate standard normal cdf.

Then the univariate case could be written $\Phi_1(x)$, to which we can arrive by starting from $\Phi_{\mu, \sigma} (x)$, then setting $\mu =0, \sigma=1$, and eliminate the zero-subscript (since absence many times essentially means "zero"). This is essentially a happy case of confounding.