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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?

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  • $\begingroup$ Where do you encounter this property? Could you please give a reference? $\endgroup$
    – Heisenberg
    Jan 10, 2019 at 8:41

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Ok, so I think I have verified the result. I guess it was a matter of defining the moment generating function of $Y\mid X$. But her it goes

$$E\left[\exp(t'Y)\right] = E\left[E\left[\exp(t'Y)\mid X\right]\right] = E\left[ \exp(t'AX + (t'Vt)/2) \right].$$ Now it follows that $$ E\left[\exp(t'Y)\right] = \exp\left( t'A\mu + (t'(A\Sigma A') + V)/2 \right), $$ since $X\sim N(\mu,\Sigma)$.

The only question left is; is it okay to define the moment generating function of $Y\mid X$ as $$E\left[\exp(t'Y)\mid X\right].$$

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