Suppose I have three random variables $x,y,z$ and they have a relation as $y=x+z$ now I have the distribution of $z \sim p(z)$, how to find $p(y|x)$. I know intuitively if I have $z\sim N(0,\sigma^2)$ then $x$ will just impact its mean and so $p(y|x)\sim N(x,\sigma^2)$.
But I want to know general mathematical procedures to find $p(y|x)$, specifically, if $z\sim p(z)$ where $p(z)$ have an undefined MGF and mean and variance both are infinite. Further, I assume that $x$ and $z$ are independent of one another.
The distribution function of $Y$ given $X$ is $F_Z(y-X)$ and the density function is $f_Z(y-X)$ where $F_Z$ and $f_Z$ are the disrtibution function and the density of $Z$ respectively.