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Let us consider $X\sim\mathcal{N}(\mu,\Sigma)$ being a $d$-dimensional multivariate Gaussian random variable. I know that it is possible to calculate the distribution of $X|S=s$, where $S$ is the sum over all components of X, and I found it is still a Gaussian. But I thought of a variation of this and I could not think how to calculate it.

Is it possible to calculate the distribution of $X|S>s$? Or more generally, given a vector $ A\in\mathbb{R}^d$, is it possible to calculate $X|A'X>s$? I could not find a result of this kind anywhere, and I wonder out of curiosity if this is known or even possible to solve analytically ...

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