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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.

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2 Answers 2

up vote 1 down vote accepted

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.

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Is $N$ large (where "large" is probably "bigger than five", in practice)? If it is, the central limit theorem gives a good approximation to the distribution of $\sum_{i=1}^N x_i,$ assuming some weak conditions on the variances (something like sum of the variances (see Feller, vol 2, page 251-252, or http://arxiv.org/pdf/math/0403375, Theorem 8)

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