I have the following question: $Let X,Y\in \mathbb{R}^n$ be jointly distributed random vectors such that: $$X\sim N_n(\mu, O), Y|X\sim N_n(X,\Sigma)$$ Find the Joint Distribution: $$X|Y=y$$
I have found the expectation $\mathbb{E}[Y] = \mu$ and variance $Var(Y) = \mathbb{E}[\Sigma] + O$ using the law of total expectation and law of total variance but I am unsure how to proceed since I do not know how to calculate $Cov(X,Y)$. Would appreciate any guidance.
Begin with the definition:
$\qquad\mathsf{Cov}(X,Y)~=\mathsf E(XY)-\mathsf E(X)\,\mathsf E(Y)$
Now apply the Law of Total Expectation.