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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.

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    Often network delays can be modeled by Gamma distributions with a common value of beta. In this case finding the distribution of the sum is easy (that is one of the reasons that they are often used). In general, you need to know the distributions, or a good approximation thereto.2011-07-01

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