Let $X$ and $Y$ be continuous random variables both having some density, not identically distributed but independent.
Imagine I'm interested in the quantile $q_{X+Y}(\alpha)$ for some $\alpha \in (0,1)$.
Does it then hold that
$$q_{X+Y}(\alpha)=\int\limits_{-\infty} ^{\infty} f_X(x)(x+q_{Y} (\alpha)) dx$$
So the idea of this expression would be the following:
Go over all possible values of $X$. If $X$ has some fixed value $x$, then the quantile of $X+Y$ is just the quantile of $Y$ plus some constant $x$.
But I might be very wrong in my intuition
