$\phi(x)$ is the prob. density. I prefer $f_X(x)$.

If you like to rewrite this in terms of a new random variable $z$ you have to use a transformation, i.e. $$f_X(x) = f_Y(y) \left| \frac{dx}{dy}\right|$$ This is valid for continuous random variables, not for discrete random variables.

You denote the prob. density by $\phi(x)$, I prefer $f_X(x)$.

If you like to rewrite this in terms of a new random variable $z$ you have to use a transformation, i.e. $$f_X(x) = f_Y(y) \left| \frac{dx}{dy}\right|$$ This is valid for continuous random variables, not for discrete random variables.