Let $Y|X=x\sim N_p(Ax+B,\Sigma)$ and $X\sim N_p(\mu,\Psi)$ verify that $$Y\sim N_p(A\mu+B,A\Psi A^T+\Sigma)$$ where $A$ and $B$ are matrix of constant terms.

I know that $$E[Y]=E[E[Y|X]]=E[AX+B]=A\mu+B$$ $$Var(Y)=E[Var(Y|X)]+Var(E[Y|X])=\Sigma+Var(AX+B)=\Sigma+A\Psi A^T$$ the law of variance total and this conditional expectation property is also valid in the multivariate case?

If the conditional distribution is normal and $X$ too, I can assume that $Y$ is normal?

I know that I could try to work with the joint distribution, but it is a lot of work. Is there another way if the above is not true?

  • $\begingroup$ I don't see how else you could do this other than multiplying $f_{Y \mid X}$ and $f_{X}$ together, rewriting the product so that you get a multivariate normal density. $\endgroup$ – Clarinetist Oct 18 '17 at 13:27
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    $\begingroup$ it really just depends on what you're able to cite as being "known" vs what you need to prove yourself. If you have to show everything, then I don't think there's any way around directly showing it like Clarinetist suggests. But if you can use as a theorem the fact that marginals of a MV gaussian are gaussian, then you just need to show $(X, Y)$ is gaussian. And etc $\endgroup$ – jld Oct 18 '17 at 14:14

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