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Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $ \Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N correlated Gaussian RVs).

I am interested in calculating the probability that $\sum_i \exp\{ x_i \} < c$ for these given parameters.

So, what is the probability that the sum of the exponential of MVN RVs will be less than a given value?
This might also be framed in terms of log-normal random variables.

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    $\begingroup$ Then the wording is wrong: one is drawing $N$ correlated univariate Gaussians. In which case the sum is a sum of log-normals. $\endgroup$
    – Xi'an
    Commented May 9, 2022 at 6:30
  • $\begingroup$ The question text is very unclear, can you please fix it? $\endgroup$
    – kjetil b halvorsen ♦
    Commented May 11, 2022 at 15:10

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