Suppose there are two random variables $X$ and $Y$, which are independent but not necessarily identical. Let $Z = X + Y$. Given a probability $\alpha$, how to find the minimal (or close to minimal) $z$ subject to $\mathbb{P}(Z > z)\geq\alpha$? It's for an embedded system, so the solution has to be both memory and computation efficient.
The question also extends to multiple random variables, say $X_1, X_2, ..., X_n$ are mutually independently and $Z = X_1 + X_2 + ... + X_n$. I truly appreciate your assistance.