# Joint gaussian conditional on its sum greater than a value

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 ...

• Could you clarify what $A$ is? Your notation suggests $A$ is a matrix but $s$ appears to be a scalar. In general $A'X$ will be a matrix Apr 25 at 9:49
• Sorry for the confusion. $A$ is a fixed vector of the same size as $X$, so $A'X$ would just be an arbitrary linear combination of each individual component of $X$ @jcken Apr 25 at 9:57
• Reparametrise into the rotated $\mathbf BX$ so that $A'X$ is the first component? Apr 25 at 10:51
• – whuber
Apr 26 at 13:10
• Thank you so much for these references!!!! @whuber Apr 26 at 17:52