I have tried to come up with an approach for the following, but failed.
Suppose $Y\mid X = x\sim N(Ax,V)$ and $X\sim N(\mu,\Sigma)$. Then $Y\sim N(A\mu, A\Sigma A^\mathsf{T}+V)$.
I can't think of results about moment generating functions which can help. Ofcourse, I can derive the mean and variance of $Y$ by iterated expectation and variance. But I can't seem to verify that $Y$ is also normal. Any suggestions?