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Let $p(x)=\mathcal{N}(x|\mu_x,\Sigma_x)$ and $p(y|x)=\mathcal{N}(y|Ax+b,\Sigma_y)$.

We want to compute $p(x,y)$. The book I'm reading says that $$ \log p(x,y) = -\frac{1}{2}(x-\mu_x)^T \Sigma_x^{-1}(x-\mu_x)-\frac{1}{2}(y-Ax-b)^T\Sigma_y^{-1}(y-Ax-b) $$ is clearly a joint Gaussian distribution, since it's a quadratic form.

However, to rewrite that expression in the form $-\frac{1}{2}(z-\mu_z)^T \Sigma_z^{-1}(z-\mu_z)$, the author ignores the linear and constant terms. The constant terms are not important because they end up in the normalization constant, but what about the linear terms?

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$\Sigma_{z}$ depends only on $A$, $\Sigma_{x}$, and $\Sigma_{y}$. It doesn't depend on $b$. However, $\mu_{z}$ does depend on $b$, $y$, and $\mu_{x}$.

You haven't told us what source you're using, so it's difficult to say much more about this.

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