I have a very general question, what is probability that a low number of independent variables together are greater than some value? Suppose you are given $X_1$, $X_2$... $X_n$, where $n$ is fairly small. How can I calculate $$P(X_1+X_2+...+X_n>a)?$$ Suppose that the mean and standard deviation are known for all $X_i$. Wouldn't I still have to known the distribution of $X_1+X_2+...+X_n$? For a large $n$ the Central Limit Theorem would imply that $X_1+X_2+...+X_n$ is normal but now it can't be used, so how do I proceed?
The exact problem that I have is: $x\sim Normal(10,3)$, $Y\sim Uniform(3,7)$ and $Z\sim Exponential (20)$. What is $P(X+Y+Z>18)$?