How do I compute the following? 


$$ \mathrm{Prob}\left( \sum\_{i=1}^N x\_i >1\right) = ?$$


where $x\_i \sim \mathrm{Log}\mathcal{-N}(\mu\_i, \sigma\_i^2)$. 


AFAIK, we do not know how the sum of log-normal distributions is distributed, so what I have in mind is to pick an appropriate interval for $x$, sample uniformly from there, and calculate sum of $x\_i$s. If it's greater than 1, then I count it as 1. Am I doing it right? Is this the "Monte Carlo Method"? Are there any (more appropriate) other methods you can suggest? I wanted to get some experts' words to be sure. Thanks. 



probability computation involving sum of log-normal random variables
The following paper: "The Distribution of Products of Beta, Gamma and Gaussian Random Variables"
Author(s): M. D. Springer and W. E. Thompson (on JSTOR)


gives a solution for the problem in the case of mean-zero independent normal random variables, using the Mellin transform.


In this case the density is given using Meijer's G-function.


