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This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?

So far I only figured out that I can do Monte Carlo or assume inequality encoded by each row of $A$ is independent for all the other inequalities; then there's an explicit formula for and upper bound using Boole inquality.

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You can determine the exact distribution of the Gaussian vector Ax but there is no closed formula for estimating its cumulative distribution function (see http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function).

You will have to estimate it numerically. Alan Genz's worked a lot about this subject it seems. I found a related question https://stackoverflow.com/questions/11109465/multivariate-normal-cdf-in-c-c-or-fortran. Implementation exists in R, Matlab and Fortran (!) at least.

Hope it helps.

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  • $\begingroup$ Sorry, I did not explain myself well. If you assume independence you get an upper bound on the result. I know for fact that you can efficiently evaluate 1D Gaussian CDF. Are there such techniques for ND gaussian? $\endgroup$
    – nivwusquorum
    Commented Mar 26, 2015 at 17:13
  • $\begingroup$ Ok, I thought you were searching for a closed form answer so my answer isn't really helpful :-). I edited my answer refering to Alan Genz's work. $\endgroup$
    – victolunik
    Commented Mar 26, 2015 at 17:29

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