# sum of independent random variables

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.

• In general you want to find the density of $Z$ and invert it. How to do this efficiently is going to depend very strongly on the specifics of the problem; what specifically are you interested in? – Michael Lugo Jul 1 '11 at 20:24
• In actuality, each random variable corresponds to the delay of a link in a network. And quantile of the delay of a path is of interest, which is the sum of delays of constituent links. Each node in the distributed network only knows the local link delays rather than path delay. – sinoTrinity Jul 1 '11 at 20:33
• 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. – deinst Jul 1 '11 at 20:55