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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 ...

kjetil b halvorsen
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Statistical Shiba inu
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  • 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 – jcken Apr 25 '22 at 09:49
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    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 – Statistical Shiba inu Apr 25 '22 at 09:57
  • Reparametrise into the rotated $\mathbf BX$ so that $A'X$ is the first component? – Xi'an Apr 25 '22 at 10:51
  • The $d+1$-variate distribution of $(X;A^\prime X)$ is Normal. Apply the usual results for conditional Normal distributions. – whuber Apr 25 '22 at 12:57
  • Could you give some more detail? My understanding on your suggestion is if I have the distribution of $(X;A'X)$, then I can calculate $(X; A'X)|A'X>s$ and marginalize the first d components? I was trying to do this without sucess – Statistical Shiba inu Apr 26 '22 at 04:57
  • My biggest issue is that I want $X|A'X>s$ and not $X|A'X=s$. Does it keep being gaussian? – Statistical Shiba inu Apr 26 '22 at 05:48
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    See, inter alia, https://stats.stackexchange.com/questions/444925, https://stats.stackexchange.com/questions/73157, https://stats.stackexchange.com/questions/163172/, https://stats.stackexchange.com/questions/281007, https://stats.stackexchange.com/questions/487234, and other hits from this search. – whuber Apr 26 '22 at 13:10
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    Thank you so much for these references!!!! @whuber – Statistical Shiba inu Apr 26 '22 at 17:52

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